Tabulation - The Tanglenomicon: Tabulation of two string tangles

6.1The First Tangle Datasets¶

This chapter will describe the methodology used to answer two of the essential questions detailed in Section 4.3.

“How do I systematically construct rational tangles?”, “How do I tell two rational tangles I make apart?”, and “How do I generate new rational tangles?”

“How do I systematically construct Montesinos tangles?”, “How do I tell two Montesinos tangles I make apart?”, and “How do I generate new Montesinos tangles?”

The methodologies outlined in this section are for relatively simple classes of tangles, the rational and Montesinos tangles. As we progress, we will see a common solution pattern that outlines the general approach to the more difficult algebraic/arborescent case in Section 6.2. That approach takes the form of the cadence:

Define the object.
Define equivalence for the object.
Identify a unique representation.
Generate those unique representatives^[1]

6.1.1Rational Tangles¶

We will see in this section that rational tangles, originally described by Conway 1, have a deeply combinatorial nature, making them among the simplest classes of tangle. This simplicity leads to the rational tangles, and their knot closures, being some of the most commonly studied objects in knot theory.

6.1.1.1Construction¶

In our development of the modified tangle calculus (Section 5.3.2) we described a way to glue tangles together, allowing us to take simple objects and build complex objects. That approach forms the basis for our construction of the rational tangles.

We start, with an intuitive description of the construction. Imagine a zero tangle (Figure 9a), now attach to that tangle a crank on the right (east) side (Figure 1). We allow, for a moment, the fixed points of the tangle to move. If we crank a half turn clockwise or anti-clockwise, we introduce a twist, if we turn the crank $n$ half turns we make an integral $n$ tangle(Section 5.3.3).

Figure 1:A set of three turns, changing a basic 0 tangle into an integral 3 tangle.

When we have completed $n$ turns we take the crank and move it to the bottom (south) side of the tangle. We turn the crank to add $n$ half turns, this time building a vertical integral tangle. Continuing this, alternating between right and bottom sets of half turns, as seen in Figure 2, a rational tangle. For a rational tangle $T$ , the list of counts for right and bottom twists is the twist vector (Definition 2) of the rational tangle.

The set of alternating turns building a rational tangle \LB3\ 2\ 1\RB. — Figure 2:The set of alternating turns building a rational tangle $\LB3\ 2\ 1\RB$ .

Formalizing this intuitive construction requires the $\vee$ and $+$ operations (Section 5.3.2), as well as the integral tangles (Section 5.3.3). This formalization was originally stated by Conway 1 but was ultimately proved by Burde and Zieschang, we will give a later construction by Kauffman and Goldman 2. For convenience, we denote a horizontal integral tangle with $a\in\Z$ crossings as $t_a$ , similarly, a vertical integral tangle with $b\in \Z$ crossings as $t_b^\prime$ .

Definition 1 (Kauffman and Goldman, Page 310 21)

To build a rational tangle, take one of the following constructions:

Start with a horizontal tangle $t_{a_0}$ . Add by $\vee$ a vertical tangle $t_{b_0}^{\prime}$ on the bottom, then add by $+$ a horizontal tangle $t_{a_1}$ on the right, then add by $\vee$ a $t_{b_1}^{\prime}$ on the bottom, and so on, stopping after a finite number of such steps at a horizontal tangle $t_{a_n}$ .
Start with a vertical tangle $t_{b_0}^\prime$ . Add by $+$ a horizontal tangle $t_{a_0}$ on the right, then add by $\vee$ a vertical tangle $t_{b_1}^\prime$ on the bottom, then add by $+$ a $t_{a_1}$ on the right, and so on, stopping after a finite number of such steps at a horizontal tangle $t_{a_n}$ .

To see the alignment between Definition 1 and our intuitive construction, view the turning of the crank as corresponding to horizontal and vertical integral tangles, and the alternating of right and bottom as corresponding to alternating $+$ and $\vee$ .

Correspondence With Extended Rational Numbers¶

Now we formally address our first two essential questions by:

Describing a notation for rational tangles.
Demonstrating a correspondence between the rational tangles and the extended rational numbers.
Showing that the correspondence distinguishes (tells apart) rational tangles.

The answer to our first essential question, “How do I systematically construct rational tangles?”, is seen by formalizing the twist vector notational strategy we saw in our intuitive formulation for rational tangles. This allows us to systematically write down a rational tangle by a list of integers.

We are now ready to answer the second of the essential questions, “How do I tell two rational tangles I make apart?” The critical observation to answer the questions is due to Conway’s use 1 of the entries of a twist vector as the entries for a continued fraction (Definition 3). Since the twist vector is of finite length, this continued fraction corresponds to an extended rational number 3, $\Q\cup\LS\infty\RS$ .

Kauffman and Goldman 2 prove (Theorem 1) that this correspondence distinguishes, tells apart, rational tangles. Meaning, given two rational tangles, if their fractions are the same the tangles are isotopic, and if the fractions are different, the tangles are not isotopic. This answers the second of our essential questions.

Observe, Theorem 1 does not discount the possibility of two non-equivalent twist vectors, those differing in at least one entry, representing the same rational tangle. Example 1 demonstrates that the possibility is true. In fact, for each rational tangle, the set of twist vectors representing it is infinite.

To effectively answer our third essential question, “How do I generate rational tangles?”, we will need to determine a unique twist vector representative for each rational tangle. A unique representative allows us to simply and efficiently write down each tangle without risk of duplicates showing up on our list.

6.1.1.2Canonical Twist Vectors¶

Identifying a unique representative will stem from properties of finite continued fractions. We start by defining a specific subclass of finite continued fractions with integer coefficients, the regular continued fractions. We frame the definition in the context of twist vectors.

Conveniently, each extended rational number corresponds to exactly two regular continued fractions 3. The first is a twist vector with the leftmost element greater than or equal to 2, and the second with leftmost element equal to 1, per (5). Observe, one of these twist vectors has an even number of entries, and one has an odd number of entries.

\begin{aligned} F\LP\LB 9\RB \RP &= \frac{9}{1}= 8+\frac{1}{1} = F\LP\LB 1\ 8\RB \RP \\ F\LP\LB 9\ 0\RB \RP &= 0+\frac{1}{9}=0+\frac{1}{8+\frac{1}{1}}= F\LP\LB 1\ 8\ 0\RB \RP \\ \end{aligned}

(5)

To identify a unique representative for a rational number and hence rational tangle, we will select, for convenience, the odd length twist vector as our unique representative.

6.1.1.3Computational Methods¶

Armed with a unique representative for a rational tangle, we can construct our computational answer to the third essential question.

Notation¶

We start by describing how we will digitally store a rational tangle. In the rational tangle case, the theoretical encoding strategy of twist vectors happens to be well suited for computational storage. A twist vector can be computationally stored identically to its written form, a list of space separated integers delimited by a pair of square braces, $\LB\ \RB$ . As we will see in Section 6.2 this direct translation of theoretical notation to computational notation is not always the case.

Generation¶

A common tactic in the knot tabulation space is to pare down the number of items that must be tabulated by leveraging symmetries of the objects being tabulated. For example, the $\LB 3\RB$ and $\LB \m 3\RB$ tangles are related to each other by a minus operation (Minus Tangle). Meaning, if we tabulate $\LB 3\RB$ , we can recover $\LB \m 3\RB$ with multiplication by $\m 1$ . This fact holds for all rational tangles as demonstrated by Lemma 1. Allowing us to focus our efforts on the rational tangle with twist vectors containing only positive entries.

We now begin our development of a generation strategy for twist canonical twist vectors of rational tangles. The method seen here utilizes a common combinatorial method for defining compositions of integers, the same is used for rational tangle counting by Bryhtan 6. Consider, for a given crossing number $n$ , what is the most “obvious” twist vector? A viable candidate for most “obvious” is a twist vector, of the form seen in (6), dropping the trailing 0 where needed.

\LB 1\ 1\ 1\ \cdots\ 1\RB

(6)

The 1’s twist vector (6) is an ideal starting point for developing a generation strategy, as it distributes the data of a rational tangle as broadly as possible. Next, we consider how we might transform the 1’s twist vector into a twist vector with a two crossing integral components. We can do this by exchanging the space between the first and second 1 with a numeric $+$ .

\begin{aligned} &\LB 1\square 1\ 1\ \cdots\ 1\RB\\\\ &\LB 1+1\ 1\ \cdots\ 1\RB\\\\ &\LB 2\ 1\ \cdots\ 1\RB\\\\ \end{aligned}

(7)

This process tells us that we can utilize the exchange of whitespace of a twist vector for $+$ to generate new twist vectors. To complete our generation, we must generate every combination of exchanges.

The 1’s twist vector with crossing number $n$ has $n$ 1s and $n-1$ spaces. In each space position, we have the option between a space and $+$ . To enumerate all $2^{n-1}$ combinations, we can simply count, in binary, from 1 to $2^{n-1}$ , as in Example 2.

Example 2 (Combinations of exchanges for

n=4

and their twist vector)

\begin{aligned} 000\to \square \square \square\to &\LB1\ 1\ 1\ 1\RB \\\\ 001\to \square \square +\to &\LB1\ 1\ 2\RB \\\\ 010\to \square + \square\to &\LB1\ 2\ 1\RB \\\\ 011\to \square + +\to &\LB1\ 3\RB \\\\ 100\to + \square \square\to &\LB2\ 1\ 1\RB \\\\ 101\to + \square +\to &\LB2\ 2\RB \\\\ 110\to + + \square\to &\LB3\ 1\RB \\\\ 111\to + + +\to &\LB4\RB \\\\ \end{aligned}

(8)

Our final refinement in this process it to transform this collection into canonical twist vectors. Half of the twist vectors, those of odd length, generated in this process are already canonical. To canonize the even length twist vectors, we append 0 to the right most position of each list, turning the even vectors into an odd canonical vectors.

We conclude the section with a set of algorithms that describe a method for computationally generating all rational tangles up to a given crossing number.

6.1.2Montesinos Tangles¶

In this section, we will use the rational tangles to build a yet more complex class of tangles, the Montesinos tangles. This building up process demonstrates one of the core strategies for tangle tabulation.

6.1.2.1Construction¶

With the rational tangles in hand, we wish to utilize that data to enumerate additional tangles. One way we have seen to build simple objects into complex objects is to combine two tangles with the $+$ or $\vee$ operation. To keep complexity under control, we start with combining tangles with repeated $+$ sum. When all summands, $R_i$ in equation (9), are rational, we call the result of the sum a Montesinos Tangle 78.

R_0+R_1+\cdots+R_n

(9)

Unique Representative¶

Next, we develop a classification of Montesinos tangles, allowing us to tell two Montesinos tangles apart. For each rational summand $R_i$ in a Montesinos tangle, we have four possibilities:

$R_i$ ’s canonical twist vector is positive and ends in 0
$R_i$ ’s canonical twist vector is positive and does not end in 0
$R_i$ ’s canonical twist vector is negative and ends in 0
$R_i$ ’s canonical twist vector is negative and does not end in 0

Observe that in the second and fourth cases, $R_i$ terminates in a horizontal integral tangle. In these cases, the tangle can be simplified by using the flype (The Flype) to move the horizontal crossings to be the right most summand, seen in Figure 3. When this process is carried out for each summand of the type in cases two and four, the resulting summands all fall into cases one and three.

We will now pare down to a single possibility, case one. Consider a summand $R_i$ in case three, meaning $\m 1 <F\LP R_i\RP<0$ except for $R_n$ , which may be integral. Theorem 1 tells us that if we can find an alternative, potentially non-canonical, twist vector that fits our needs, we are free to exchange without impacting topology. What we would like is an odd length twist vector, where every entry is positive, except for the rightmost, which is a negative value, see Figure 4.

On the left rational tangle \LB \m1\ \m2\ \m1\ \m1\ 0\RB and on the right
\LB 3\ 2\ \m1\RB. These tangles have fractions
F\LP\LB \m1\ \m2\ \m1\ \m1\ 0\RB\RP=\frac{\m4}{7} and
F\LP\LB 3\ 2\ \m1\RB\RP=\frac{\m4}{7} showing the tangles to be isotopic. — Figure 4:On the left rational tangle $\LB \m1\ \m2\ \m1\ \m1\ 0\RB$ and on the right $\LB 3\ 2\ \m1\RB$ . These tangles have fractions $F\LP\LB \m1\ \m2\ \m1\ \m1\ 0\RB\RP=\frac{\m4}{7}$ and $F\LP\LB 3\ 2\ \m1\RB\RP=\frac{\m4}{7}$ showing the tangles to be isotopic.

This ensures that the fraction is still negative but will allow us to flype the terminal horizontal integral tangle to the right. Rockett and Szüusz give a lemma that establishes the existence of such a twist vector for each rational number.

We have now shown that each case (2,3,4) can be transformed into the first case, and we can define a canonical form for Montesinos tangles.

6.1.2.2Computational Methods¶

Notation¶

Before we can generate Montesinos tangles, we need to define an efficient notation for computation and storage. Similar to what we saw in the rational tangle case, the theoretical notation for Montesinos tangles is sufficient for computation. However, with eyes on future computational work, we will generalize our notation to increase reusability and the efficiency of storage.

Montesinos tangles are simple forms of the algebraic tangles (Definition 7), so we will build a notational strategy for general algebraic tangles. The strategy seen here is similar to those found in Conolly, Caudron , and Gren, Sulkowska, and, Gabrovšek 91011. The theoretical notation for algebraic tangles is outlined in Section 5.3.2 and seen in Example 3. We can simplify the notation, without losing fidelity, by substituting the integral leaf tangles for rational tangle twist vectors. Additionally, we can improve the storage overhead by storing the tree as a string in Polish notation 12. Storing in Polish notation allows us to drop all the parentheses from our notation, saving two bytes in each instance.

Generation¶

In this section, we will design an algorithm that allows us to efficiently generate new Montesinos tangles up to a given crossing number. As we saw, the construction of a Montesinos tangle is based on the repeated summation of rational tangles. Consequently, our generation strategy will utilize our table of rational tangles.

To start, we need a mechanism that allows us to select all possible combinations of rational tangles with crossing numbers that sum to our target. For a Montesinos tangle $T$ , the set of rational components $\LS R_i\RS_i^n$ combined with the integral component $k$ corresponds to an ordered list of crossing numbers as in (10).

CN\LP R_0\RP,\ \cdots,\ CN\LP R_n\RP,\ CN\LP k\RP

(10)

We call a list of the form seen in (10) a stencil for a Montesinos tangle. By construction, every canonical Montesinos tangle relates to precisely one stencil. Observe, that $2\leq CN\LP R_i\RP$ since $\frac{1}{2}$ is the lowest crossing number rational tangle with fraction in the unit interval.

Generation for all Montesinos tangles of a given crossing number at this point can be broken down into two steps:

Generate all stencils
Fill in the stencils with all rational tangles of the appropriate crossing number whose fraction is in the unit interval (plus an integral tangle in the rightmost position)

For the first step, we require a mechanism for breaking an integer into all possible combinations where the parts sum to the integer. Luckily, we have already seen how to do this, in the context of a twist vector. We follow the same counting algorithm outlined in Algorithm 1 however, we modify the algorithm to keep both the even and odd outputs, but filter out stencils with entries less than two.

Example 4

The set of all possible stencils for crossing number 5:


$1\ 1\ 1\ 1\ 1$	$2\ 1\ 1\ 1$	$1\ 2\ 1\ 1$	$1\ 1\ 2\ 1$
$1\ 1\ 1\ 2$	$3\ 1\ 1$	$1\ 3\ 1$	$1\ 1\ 3$
$2\ 2\ 1$	$2\ 1\ 2$	$1\ 2\ 2$	$3\ 2$
$2\ 3$	$4\ 1$	$1\ 4$	$5$

$\!\,$ The set of stencils for crossing number 5:


$3\ 2$	$2\ 3$

For the second step, for each entry in the stencil we create a list of rational tangles in the unit interval with that entry for a crossing number. For the rightmost entry of the stencil we also include the horizontal integral tangles with crossing number equal entry. This creates all combinations of input tangles given by the stencil.

Example 5

The set of rational tangles of crossing numbers two and three:

Two	Three
$[1\ 1\ 0]$	$[2\ 1\ 0]$
$[2]$	$[1\ 2\ 0]$
	$[3]$

The set of stencils for crossing number five, with rational tangles inserted :

$3\ 2$	$2\ 3$
$+[2\ 1\ 0][1\ 1\ 0]$	$+[1\ 1\ 0][2\ 1\ 0]$
$+[1\ 2\ 0][1\ 1\ 0]$	$+[1\ 1\ 0][1\ 2\ 0]$

We conclude the section with a set of algorithms that describe this method for computationally generating all Montesinos tangles up to a given crossing number.

6.2Arborescent Tangles¶

This section describes the methodology we use to answer the final of the essential questions detailed in Section 4.3.

“How do I systematically construct algebraic/arborescent tangles?”, “How do I tell two algebraic/arborescent tangles I make apart?”, and “How many algebraic/arborescent tangles can I create?”

In this thesis so far we have worked with the algebraic tangles (Definition 7) constructed with Conway’s tangle arithmetic (Operations on Tangles).

In this section we will leverage a slightly different, but equivalent, construction given by Bonahon and Siebenmann 8 the arborescent tangles. The one-to-one correspondence between the classes will become clear as we introduce the construction for arborescent tangles.

This section starts with an overview of Bonahon and Siebenmann’s 8 definition of arborescent knots and tangles (Section 6.2.1). We then give their smooth and combinatorial constructions of arborescent knots and tangles (Section 6.2.2). Next, we give original work extending Bonahon and Siebenmann’s canonical construction to a local view (Section 6.2.2.4). This local view is leveraged to define a unique representative for each class of arborescent tangles (Section 6.2.3). Finally, we will describe an original algorithm and notation that directly enumerates those unique representatives (Section 6.2.4).

6.2.1Definition of Arborescent¶

We now give a high-level description of the manifold theory underpinning the theory of arborescent knots and tangles. A full treatment of the manifold theory can be found in Bonahon and Siebenmann 8. Our first concept is that of a knot pair, which serves as the underlying structure for all the smooth objects in this subsection.

The arborescent vignette from seen with Conway spheres. — (a)The arborescent vignette from Figure 5 seen with Conway spheres.

The arborescent vignette showing a 1 crossing tangle to be arborescent. — (a)The arborescent vignette from Figure 5 seen with Conway spheres.

Observe that arborescent knots are characterized by a collection of Conway spheres (circles). Choosing to not fill one, or more, of these Conway spheres yields a tangle.

We see from the above the first portion of the correspondence between the algebraic and arborescent tangles. Each algebraic tangle can be naturally decomposed into a collection of nested arborescent knot vignettes given by its operations $+$ and $\vee$ .

6.2.2Weighted Planar Trees¶

6.2.2.1Construction of Arborescent Knots from Weighted Planar Trees¶

This subsection begins with an introduction to Bonahon and Siebenmann’s 8 construction of arborescent knots and tangles in the smooth setting. This is followed by the development of a combinatorial representation for arborescent knots and tangles 8. We deviate slightly from Bonahon and Siebenmann’s introduction but ultimately arrive at the same structure. In our introduction we develop partial solutions, then progressively modify those partial solutions until they fit our needs. Next, we describe Bonahon and Siebenmann’s 8 operations on the combinatorial structure, which allow us to systematically modify the structure without changing the topology. This subsection finishes with the classification of arborescent knots and tangles given by Bonahon and Siebenmann 8 as well as our extension from a global to local viewpoint.

Bands and Plumbing Squares¶

Our first step in describing a notation for the arborescent knots 8 is describing a plumbing operation on bands. A band with a plumbing square is a band $S^1\times\LB 0,1\RB$ , along with an oriented square on the band such that two of the sides of the square intersect the boundary of the band. Two examples of bands with plumbing squares can be seen in Figure 7.

(a)A band with a plumbing square facing the viewer.

A band with the plumping square facing away from the viewer. We are looking through the band. — (a)A band with a plumbing square facing the viewer.

Plumbing bands¶

We now glue the bands seen in Figure 7 together with an operation called plumbing. Consider the orientation given in the green band’s plumbing square. We will call the blue arrow $X$ and the thicker red arrow $Y$ ; similarly for the blue band with $X^\prime$ and $Y^\prime$ . We plumb the bands together along their plumbing squares, with the requirement that the orientation labels are mapped $X\to Y^\prime$ and $Y\to X^\prime$ . Finally, we forget the boundaries of the plumbing squares, leaving only the joined boundaries of the bands. The result of plumbing as well as a local picture for plumbing can be seen in Figure 8.

Our plumbing band construction can be turned into a knot by adding a series of half-twists to our plumbing bands (Figure 9a and Figure 9b). When forming the half twists, we have two options for direction relative to the band, we call one positive (left handed twists) and one negative (right handed twists). We note that the twists appear in unique regions of the band, determined by their position relative to the plumbing squares.

Band with two negative half twists and three plumbing squares. — (a)Band with two negative half twists
and three plumbing squares.

Band with three positive half twists and one plumbing square. — (a)Band with two negative half twists
and three plumbing squares.

Successive plumbing yields collections of bands like those seen in Figure 10a. Finally, turning Figure 10a into a knot is as simple as removing the interior of the band, leaving only the boundary, per Figure 10b.

It is important to note that, for creating arborescent knots, we must restrict plumbing from creating “cycles” of bands. That is a chain of plumbing beginning and ending with the same band, as seen in Figure 11. If we allow cycles in the bands, we may create a polygonal tangle, defined in Tangle insertions. These polygonal tangles contain portions that do not satisfy Definition 8, so are not arborescent presentations.

A collection of bands plumbed in such a way that the last band is plumbed to
the first band in a cycle. — Figure 11:A collection of bands plumbed in such a way that the last band is plumbed to the first band in a cycle.

We now establish some language for describing the relative positions of bands. This language will be reused when we transition to the combinatorial setting and is widely used in graph theoretic settings.

We claim the plumbing band construction is in correspondence with the definition of arborescent seen in Definition 7. To see this, we take each plumbing band and encapsulate it in a $S^2$ so that the corners of the plumbing squares lie on the $S^2$ , giving us the vignette seen in Figure 6a.

Weighted Planar Trees¶

The band construction we have developed for arborescent knots, as it stands, is completely unsuited for machine computation. In this subsection, we lay out a line of reasoning leading to a combinatorial encoding strategy 8 for the plumbing band construction of arborescent knots and tangles. The line of reasoning starts by presenting the required data of arborescent knots and tangles that any combinatorial representation must encode. We then propose partial solutions, each progressively closer to the full encoding described by Bonahon and Siebenmann 8. As we will see, the encoding strategy ultimately takes the form of a modified rooted plane tree, a specialized flavor of graph theoretic tree.

Inventing a combinatorial encoding strategy means we first have to identify the essential information that is needed to construct arborescent knots and tangles from band plumbings. The two essential pieces of information that must be encoded by any combinatorial strategy for notating plumbing of bands are the following:

The parent child relationship between bands
The twists on bands, and their positions relative to the band’s children (plumbing squares)

Explicit details expanding on why these two pieces of information are essential can be found in Bonahon and Siebenmann 8. We will see in the following subsections ways in which these data are essential, albeit in specialized cases.

Consider the first piece of essential information our combinatorial strategy must encode, the parent child relationships between bands. Perhaps the most commonly computationally utilized structure that encodes relational data is an abstract graph. We imagine how an abstract graph might be used for encoding the relationship of bands. One solution is to map bands to vertices and plumbing relationships to edges. In the discussion of the band construction, we restricted plumbing from forming cycles (Figure 11). A result of this restriction is that any abstract graph must also have no cycles, meaning all the graphs we will work with, abstract or otherwise, are actually trees. We will call the data of a vertex and a collection of bonds (half-edges) associated with plumbing squares the local picture around a vertex.

We have partially completed our goal of encoding the essential information of band plumbing in a combinatorial object. Unfortunately, an abstract tree doesn’t encode all the essential data. Particularly, our second piece of information, the positions of children and weights, is not easily seen in an abstract tree. To solve this problem, we will instead use a modified version of an abstract tree, a rooted plane tree, for our encoding.

In a rooted plane tree $\Gamma$ , at each vertex $v$ , the children of $v$ have an ordering inherited from the total order of $\Gamma$ , we call this ordering of the children the cyclic order of the children. The cyclic order gives us two natural ways to draw $v$ and its children in the plane. We may choose to draw the children anti-clockwise in one of increasing or decreasing order of index. The realization of these two options can be seen in Figure 12. A universal choice of increasing or decreasing yields a unique realization of a rooted plane tree in the plane.

(a)The local picture of a vertex with child labels increasing in anti-clockwise order.

The local picture of a vertex with child labels decreasing in anti-clockwise order. — (a)The local picture of a vertex with child labels increasing in anti-clockwise order.

Indexing the total order of a tree¶

We will now describe an ideal indexing for a rooted plane tree.

(a)A rooted plane tree with ideal indexing. The index of each vertex is seen inside the vertex.

A rooted plane tree with indexing that is not ideal. The index of each vertex is seen inside the vertex. — (a)A rooted plane tree with ideal indexing. The index of each vertex is seen inside the vertex.

The final data we need to record is the position and count of half twists relative to plumbing squares. We observed earlier that the half twists on a band must lie in a unique region determined by position relative to plumbing squares on a band. This relationship can be recreated for a rooted plane tree by annotating the local view of a vertex with an integer placed in the regions between bonds. The relationship between a plumbing band and a weighted vertex in a rooted plane tree can be seen in Figure 14a. The weights placed in regions between bonds inherit a cyclic order from the cyclic order of the bonds. Each weight falls in the region between two bonds. We assign to each weight (including zero weights) the lower of the two indices. This aligns with assigning to the weight the index that appears before it in the anti-clockwise planar realization of the cyclic order, per Figure 14b.

(a)The local view of a vertex with the weights two, zero, and zero.

The local view of a vertex with weight. Notice the index of the weights comes from the bond
“before” it in the planar realized cyclic order. — (a)The local view of a vertex with the weights two, zero, and zero.

We can see a full example of a tree with its associated plumbed construction in Figure 15a. We call this fully realized combinatorial recipe for an arborescent knot a weighted planar tree.

(a)The tree describing the plumbing of bands. Each vertex represents the band illustrated near it.

The realization by plumbing bands of the tree in — (a)The tree describing the plumbing of bands. Each vertex represents the band illustrated near it.

Weighted Planar Tangle Trees¶

Our construction to this point has been concerned with the notation for knots and links. We now give a modification of this notation for tangles. A weighted planar tree, as in Figure 17a, can be modified to represent a tangle by allowing a free bond (half-edge) to be attached to a vertex, that is, to allow bands to have a non-plumbed plumbing square. We realize the non-plumbed square as a Conway circle for a two string tangle as in Figure 17b. To consistently orient the Conway sphere’s interior, we align the north boundary points of the Conway sphere with the top (up in the band orientation) boundary component and place the NW corner first (following the orientation of the boundary), per Figure 16. Plumbing two bands then corresponds to the action of gluing a pair of tangles together on their Conway spheres so that boundary points align.

Figure 16:The orientation of a Conway sphere is given by a plumbing square on a band of an arborescent tangle. The orientation of the underlying plumbing square is shown. This aligns with a left hand rule with $Y$ the thumb, $X$ the index finger, and $Z$ middle finger, with $Z$ pointing away from the center of the band, out of the page in this case.

A tree may have many free bonds, with each free bond representing a unique boundary Conway sphere. Each boundary component serves as a location where a tangle can be inserted to form a knot or link. For our efforts in enumerating two string tangles, we restrict our focus to trees that have a single free bond. In tangle trees with a single free bond, we designate the vertex with the free bond as the root of the tree.

(a)The plumbing realization of an arborescent tangle.

With an isotopy of the tangle and inversion of the Conway circle given by the
non-plumbed square we have the realization of as a
traditional orthogonally projected tangle. — (a)The plumbing realization of an arborescent tangle.

We will see that keeping track of the location of the fixed points of the boundary sphere is important when determining tangle equivalence. This is due to the need to maintain the rational number (Definition 3) associated with the “rational tangle” subtangles of a tree, prompting us to assign rotation information to the free bonds. This information takes the form of labels from the members of $V_4$ of the Klein four-group $\iota,\xi,\zeta,\eta$ . Each of these labels corresponds to a rotation of the Conway sphere around an axis in $\R^3$ , as seen in Figure 18 and Figure 19. Full details for the manifold theory underpinning these markings are found in Bonahon and Siebenmann 8. We call such a labeled tree a weighted planar tangle tree.

The effect of the V_4 rotations on each of the — (a)The identity rotation (no rotation)

\iota for no rotation — (a) $\iota$ for no rotation

\xi rotates around the X-axis — (a) $\iota$ for no rotation

6.2.2.2Anatomy of a tree¶

In this subsection, we will describe several portions of weighted planar trees: the ring subtree, essential vertex, and the sticks of a tree.

Ring subtree¶

We will now describe the ring subtrees of a weighted planar tree, which locally appear as Figure 20.

Now, resolving the plumbing for the positive subtree, we arrive at bands as in Figure 21.

Notice that the boundary of these plumbed bands has three components, as seen in Figure 22.

With an isotopy of the tangle and inversion of the Conway circle given by the non-plumbed square, we can arrange our plumbed bands into the standard tangle projection seen in Figure 23. This tangle projection tells us that the subtree in Figure 21 is, depending on location of $\text{NW}$ , either the zero or infinity tangle with a ring.

Essential Vertex¶

We now classify each vertex into one of two classes, the essential vertices and the non-essential vertices.

As an example, consider the vertices seen in Figure 24.

A weighted planar tree annotated with essential vertices in orange and
non-essential in blue — Figure 24:A weighted planar tree annotated with essential vertices in orange and non-essential in blue

Sticks of a Tree¶

The final part of the anatomy of a tree we will consider is the sticks of a tree.

As an example, consider the tree seen in Figure 24, the sticks of which can be seen in Figure 25.

Sticks of the tree from , six half-open sticks and one open
stick. — Figure 25:Sticks of the tree from Figure 24, six half-open sticks and one open stick.

By construction, a stick subtree of $\Gamma$ may have 0, 1, or 2 free bonds (seen in Figure 26). We call a stick with 0 free bonds closed, 1 free bond half-open, and 2 free bonds open. Additionally, we call a stick where each vertex has a single weight a proper stick, and we call a vertex on the end of a stick an end vertex.

From top to bottom, a closed, half-open, and an open stick. Each end vertex
is colored in red. — Figure 26:From top to bottom, a closed, half-open, and an open stick. Each end vertex is colored in red.

Integral Tangles¶

When a weighted planar tangle tree is a half-open stick containing a single vertex with a single weight $w_0$ we call it an integral tangle tree.

Figure 27:A stick realized as an integral tangle.

Rational Tangles¶

Bonahon and Siebenmann 8 give a correspondence between stick tangle trees (a stick with a single free bond) and Conway’s rational tangles 1. An example of the correspondence can be seen in Figure 28.

Figure 28:A stick tangle tree realized as a rational tangle.

Tree Crossing Number¶

Finally, we define the Tree Crossing Number (TCN) of a weighted planar tangle tree. This corresponds to the crossing number of the tangle diagram given by the weighted planar tangle tree.

6.2.2.3Calculus of Weighted Planar Trees¶

This subsection develops a set of moves, $F^\prime_3,\ F_2,\ F_1$ , and $R^\pm$ , on the weighted planar trees described in Weighted Planar Tangle Trees. We will restrict our discussion to a subset of the calculus of weighted planar trees. A full description of the calculus can be found in Bonahon and Siebenmann 8. The $F^\prime_3,\ F_2,\ F_1,\text{ and}\ R_\pm$ moves allow us to systematically modify, without changing the topology, a weighted planar tree and form the basis for the classification of the arborescent knots.

The $F^\prime_3$ Move¶

The first move, and as we will see, the most important in distinguishing tangles, is the $F_3^\prime$ move. In this move we consider the local picture of a vertex. In the local view, isolate a single bond (half-edge corresponding to a plumbing square of a band), then move a weight across that bond. The impact of this movement of a weight propagates to the descendants of the subtree attached to the bond (plumbing square) but leaves unchanged all other weights and bonds (including their attached subtrees) in the local view of the object vertex.

Definition 18 (Bonahon and Siebenmann, Section 12.7.3 8)

The $F_3^\prime$ move on a weighted arborescent tree moves a weight $W$ as in Figure 29 and, if $W$ is odd, reverses the cyclic order of weights and bonds at all vertices of the subtree attached to the purple bond (half-edge) lying at odd distance (count of edges between two vertices) from the vertex shown. Also, when $W$ is odd, apply $\xi$ ( $X$ -axis rotation Figure 19) to all free bonds in the subtree attached to the purple bond (half-edge) that are attached to a vertex at even distance from the vertex shown, and $\eta$ ( $Y$ -axis rotation Figure 19c) to those at odd distance. The rotations are relative to the local orientations of the plumbing squares on the bands corresponding to vertices at odd distance from the vertex carrying weight $W$ .

$An F_3^\prime move on a subtree. The purple bond indicates the subtree impacted by the move. The blue portion of the local view indicates all other bonds and weights of the vertex.$

Figure 29:An $F_3^\prime$ move on a subtree. The purple bond indicates the subtree impacted by the move. The blue portion of the local view indicates all other bonds and weights of the vertex.

The $F_3^\prime$ move is a derivative of the more general $F_3$ move described in Bonahon and Siebenmann 8.

Definition 19 (Bonahon and Siebenmann, Section 12.7.1 8)

The $F_3$ move replaces the left side of Figure 30 with the right side, where the cyclic order of bonds and weights is reversed at all vertices in the subtree attached to the purple bond (half-edge) of the vertex shown and at odd distances from this vertex. Also, apply $\xi$ ( $X$ -axis rotation) to all free bonds in the subtree attached to the purple bond (half-edge) that are attached to a vertex at even distance from the vertex shown, and $\eta$ ( $Y$ -axis rotation) to those at odd distance. The rotations are relative to the orientation of the plumbing square (Conway sphere) of the the band the weight is moving across, per Figure 16.

Figure 30:An $F_3$ move on a subtree. The purple bond indicates the subtree impacted by the move. The blue portion of the local view indicates all other bonds and weights of the vertex.

We will now consider some examples of the $F_3^\prime$ move. While we explore these examples, we view $F_3^\prime$ from the perspective of an object vertex (object band). The object vertex may have one or more children that we will act on with $F^\prime_3$ . When we translate $F_3$ and $F_3^\prime$ into practice we are free to operate on children as well as the parent of a vertex (band).

$F_3^\prime$ on bands¶

The translation of crossings across child bands models the traditional flype move of “Tait Flyping Conjecture” 13 fame. To see the correspondence between $F_3^\prime$ and flype we need to view the plumbed child (or parent) band as a tangle, $T$ . We can then carry out $F_3^\prime$ over this tangle. Tracking the parts of this operation, we can see the correspondence in Figure 31.

Flype and F_3^\prime, with orientation of the Conway sphere given by — Figure 31:Flype and $F_3^\prime$ , with orientation of the Conway sphere given by Weighted Planar Tangle Trees

When this is carried out, for an odd number of crossings, the child band is inverted so it lies inside the parent band (Figure 32). The inversion reverses the cyclic order of the child, as described in Definition 18.

The odd F_3^\prime case on a band model realization of the given portion of a tree, with local
X-axis in blue and Y-axis in red given relative to the purple band. Yielding a \xi (X-axis
rotation) to all bands plumbed to the purple band or plumbed at even distance (counting plumbing
squares) from the orange band, and \eta (Y-axis rotation) to the purple band and those plumbed
at odd distance from the orange band. Note that the orientations of the plumbing squares must agree
before and after F_3^\prime. Following the orientations with the left hand
(Figure~\ref{wpt-construc-fig-band_orientation}) rule shows the orientation of the purple band
reverses in X and Z in the second image due to the rotation in Y (we are no longer looking
through the band in the second image). — Figure 32:The odd $F_3^\prime$ case on a band model realization of the given portion of a tree, with local $X$ -axis in blue and $Y$ -axis in red given relative to the purple band. Yielding a $\xi$ ( $X$ -axis rotation) to all bands plumbed to the purple band or plumbed at even distance (counting plumbing squares) from the orange band, and $\eta$ ( $Y$ -axis rotation) to the purple band and those plumbed at odd distance from the orange band. Note that the orientations of the plumbing squares must agree before and after $F_3^\prime$ . Following the orientations with the left hand (Figure~\ref{wpt-construc-fig-band_orientation}) rule shows the orientation of the purple band reverses in $X$ and $Z$ in the second image due to the rotation in $Y$ (we are no longer looking through the band in the second image).

Applying $F_3^\prime$ to an even number of crossings is equivalent to applying the move on two sets of an odd number of crossings. When the first set of odd crossings is applied, the child band is inverted; the second set inverts the child again, leaving it where it began (Figure 33).

The even case of the F_3^\prime
with the purple band and any descendants of the purple band
remaining unchanged. — Figure 33:The even case of the $F_3^\prime$ with the purple band and any descendants of the purple band remaining unchanged.

We expand to an example where the child band has descendants, as in Figure 34. Observe that when the $F_3^\prime$ is applied in this case, the child band and every band even distance from it (odd distance from the parent) are inverted.

$F_3^\prime when applied to a band (gray) with a child (orange) and grandchild (light blue). We read the bands following the local orientation of the plumbing squares. Before the move is applied, the child (orange) band is traversed as; the blue band, a green star, a green circle, and back to the parent. After F_3^\prime it is traversed as; a green circle, a green star, the blue band, and back to the parent. The blue band is traversed as; yellow star, yellow circle, and back to orange band.$

Figure 34: $F_3^\prime$ when applied to a band (gray) with a child (orange) and grandchild (light blue). We read the bands following the local orientation of the plumbing squares. Before the move is applied, the child (orange) band is traversed as; the blue band, a green star, a green circle, and back to the parent. After $F_3^\prime$ it is traversed as; a green circle, a green star, the blue band, and back to the parent. The blue band is traversed as; yellow star, yellow circle, and back to orange band.

$F_3^\prime$ Examples¶

Consider the weighted planar tree in Figure 35 and Figure 36, the left trees in each agree in all but a weight of a single vertex, what we will call our object vertex which is marked in orange. The weight of this object vertex has been changed from -2 in Figure 35 to -3 in Figure 36.

We will first walk through Figure 35. In this example, our object weight is even, applying $F_3^\prime$ to the tree, the impacted subtree (purple subtree in Figure 29) is unchanged except for free bonds, which are altered as described in Definition 18.

F_3^\prime on a weighted planar tree with even weight. The weight of the object weight is even, so
the impacted subtree is unchanged. — Figure 35: $F_3^\prime$ on a weighted planar tree with even weight. The weight of the object weight is even, so the impacted subtree is unchanged.

In Figure 36, the object weight is odd, applying $F_3^\prime$ the cyclic order of vertices of the impacted subtree at an odd distance are reversed. Additionally, all free bonds in the impacted subtree are altered as described in Definition 18.

F_3^\prime on a
weighted planar tree with odd weight clockwise. Note the
changes in the relative positions of subtrees after the application of
F_3^\prime. — Figure 36: $F_3^\prime$ on a weighted planar tree with odd weight clockwise. Note the changes in the relative positions of subtrees after the application of $F_3^\prime$ .

The $F_2$ Move¶

Our second move, $F_2$ , is a special application of the general $F_3$ move.

$F_2$ is equivalent to applying $F_3$ to a vertex by moving a $\pm 1$ weight around a full cycle of the children (and parent), as in Figure 37a. If the vertex has no weights, the zero weight is split into +1 and -1, one of which completes the cycle. The +1 and -1 then cancel, returning the vertex to zero weight. The result of carrying out $F_2$ on a weighted planar tree can be seen in Figure 37b.

F_3 moving a weight in a complete cycle — (a) $F_3$ moving a weight in a complete cycle

F_2 on a weighted planar tree. Observe the changes to the entire tree, as
opposed to the changes of F_3^\prime which impact only a subtree. — (a) $F_3$ moving a weight in a complete cycle

Observe that vertices can be partitioned into two equivalence classes. Those changed by $F_2$ applied at an even distance from the root, and those changed by $F_2$ applied at an odd distance from the root. We write $F_2$ on the even class as $F_{2e}$ and odd as $F_{2o}$ .

The $F_1$ Move¶

The third of the $F$ moves is the $F_1$ move, which is a repeated application of the $F_2$ move.

To realize $F_1$ as $F_2$ moves, we successively apply $F_{2e}$ and then $F_{2o}$ to the tree. Observe that the combination of $F_{2e}$ and $F_{2o}$ modifies the free bonds by $\xi\eta=\zeta$ .

The $R^\pm$ moves¶

The $R$ move, or ring move is the final move we will describe on weighted planar tangle trees and deals with the ring subtrees of a tree. The result of a ring move on a tangle can be seen in Figure 38.

R^- on on a tangle representation of a tree. — Figure 38: $R^-$ on on a tangle representation of a tree.

6.2.2.4Canonical Weighted Planar Tangle Trees (CWPTT)¶

Observe that the weighted planar tangle trees we have seen are badly non-unique representatives for arborescent tangles. The first step to finding a unique preferred representative is to put some additional conditions on a weighted planar tree, $\Gamma$ . The following conditions pare down the equivalence class of an arborescent tangle to a more manageable level.

Definition 24 (Bonahon and Siebenmann, Section 12.8.2 8)

A weighted planar tree is called a canonical weighted planar tangle tree (CWPTT) if it has a single free bond with a label from $V_4$ and satisfies the following conditions:

Weight Condition (W) At each vertex of $\Gamma$ , at most one weight is non-zero.
Stick Conditions
1. (S.0) On any stick the weights of the vertices are non-zero except for end vertices that have a bond free in $\Gamma$ and for the case $\Gamma$ is Figure 41a or Figure 41b.
(a)The zero tangle.
(b)The infinity tangle.
2.(S.A) The non-zero weights along any stick are of alternating sign. 3. (S.1) No end vertex of a stick has weight $\pm 1$ unless it has a bond free in $\Gamma$ .
One of:
- Positivity Condition (P) Except for those with a free bond, there are no sticks in $\Gamma$ of the forms Figure 42a or Figure 42b.
  (a)The -2 integral tangle.
  (b)A fully open stick with -2 crossings.
- Negativity Condition (N) Except for those with a free bond, there are no sticks in $\Gamma$ of the forms Figure 43a or Figure 43b.
  (a)The 2 integral tangle.
  (b)A fully open stick with 2 crossings.

Bonahon and Siebenmann show in Corollary 1 that these conditions are sufficient to realize every arborescent tangle. In fact, every weighted planar tangle tree can be turned into a CWPTT by a series of moves in an extended calculus on weighted planar trees 8. We call this process canonization of a weighted planar tangle tree.

We note some consequences of the positivity and negativity condition. First, a positive CWPTT ( $\LP +\RP$ -CWPTT) can be transformed by a sequence of moves in the extended calculus of weighted planar trees into a negative CWPTT ( $\LP -\RP$ -CWPTT). Similarly, a negative CWPTT can be transformed into a positive CWPTT. Second, we note that a CWPTT, with no modification, can be both positive and negative. We will refer to these trees as neutral trees.

Bonahon and Siebenmann give a classification of arborescent tangles via moves on CWPTT.

Further, Bonahon and Siebenmann describe an algorithm for producing these sequences of moves. This algorithm will be useful to us in Section 6.2.4.2.

Canonical Vertex¶

The conditions for a weighted planar tree to be a CWPTT are phrased for the global context of a weighted planar tangle tree. We now recontextualize those conditions for a local view, a single vertex of the tree.

Definition 25

A vertex $v_i$ of a weighted planar tangle tree $\Gamma$ with a single free bond labeled from $V_4$ is said to be $\LP+\RP$ -canonical if $v_i$ has at most one non-zero weight $w_i$ , and $i$ is zero (the root) with no conditions or $i$ is not zero with the following conditions satisfied:

If the valence of $v_i$ is 1 and all of:
- $w_i\neq 0$ unless $i=1$ and $w_0=0$ (the weight of the root)
- $w_i\neq \pm 1$
- If the valence of $v_{i-1}$ (the parent) is 2 then $\text{sign}\LP w_i\RP\neq\text{sign}\LP w_{i-1}\RP$ unless $i=1$ and $w_0=0$
- If the valence of $v_{i-1}$ (the parent) is greater than 2 then $w_i\neq -2$
If the valence of $v_i$ is 2 and all of:
- $w_i\neq 0$
- If valence of the parent or valence of the child is greater than 2 (is essential) then $w_i\neq \pm1$
- If valence of the parent and valence of the child is greater than 2 (both are essential) then $w_i\neq \m2$
- If valence of the parent is 2 then $\text{sign}\LP w_i\RP\neq\text{sign}\LP w_{i-1}\RP$ (the parent)
- If valence of the child is 2 then $\text{sign}\LP w_i\RP\neq\text{sign}\LP w_{i+1}\RP$ (the child)

From this definition, we now show that these conditions are identical to those in the global context of Definition 24.

The definition and proof for $\LP -\RP$ -canonical vertices are identical. Similarly to the canonical tree case, we define a third positivity class for a vertex, the neutral vertex, a vertex that is both $\LP -\RP$ -canonical and $\LP+\RP$ -canonical.

6.2.2.5Minimalization of CWPTT¶

CWPTT are Not Minimal¶

A common measure for the complexity of knots and their relatives is the minimal crossing number. That being the least number of crossings needed to realize the object in a diagram, we call that diagram the minimal diagram. It is natural to ask if our CWPTT are minimal representatives among either all representations or arborescent representations. A quick analysis of the process of canonization demonstrates that CWPTT are unfortunately far from minimal even among arborescent representatives. An example of canonizing a tangle, making that tangle non-minimal, is seen in Figure 44a.

A minimal presentation of a arborescent tangle in both its orthogonal projection
as well as its weighted planar tree. — (a)A minimal presentation of a arborescent tangle in both its orthogonal projection as well as its weighted planar tree.

A non-minimal presentation of the same arborescent tangle as
in both its orthogonal projection as well as its CWPTT. — (a)A minimal presentation of a arborescent tangle in both its orthogonal projection as well as its weighted planar tree.

Canonization Can Increase Complexity¶

As we have seen, a CWPTT often does not realize a minimal crossing representative for an arborescent tangle. Since minimal crossing number is such a common measure for complexity, we should understand how canonization impacts the crossing number complexity of a CWPTT. We will accomplish this by identifying a (non-unique) minimal arborescent representative for each tangle. That is, a weighted planar tangle tree with minimal TCN among all weighted planar tangle trees in its equivalence class. To begin we expand our understanding of the moves in the calculus of arborescent tangles to those that alter weights arithmetically. These moves are related to the arithmetic operations on continued fractions 8.

Definition 26 (Bonahon and Siebenmann, Section 12.3 8)

When carrying out the following arithmetic moves the relative positions of weights are critical to the invariance of the underlying knot pair.

(0.1) The 0.1 move replaces the left side with the right side of Figure 45.
Figure 45:Move 0.1 on a weighted planar tree.
(0.2) The 0.2 move replaces the left side with the right side of Figure 46. Additionally, the cyclic order of all descendants in the purple subtree is reversed and $\zeta$ ( $Z$ -axis rotation Figure 19d). The vertices $a$ and $b$ need not be valence two, either or both may have a valence greater than two.
Figure 46:Move 0.2 on a weighted planar tree.
(1.1) The 1.1 move replaces the left side with the right side of Figure 47.
Figure 47:Move 1.1 on a weighted planar tree.
(1.2) The 1.2 move replaces the left side with the right side of Figure 48. The vertices $a$ and $b$ need not be valence two, either or both may have a valence greater than two.
Figure 48:Move 1.2 on a weighted planar tree.
(2.1) Replace the left side with the right side of Figure 49
Figure 49:Move 2.1 on a weighted planar tree.
(2.2) Replace the left side with the right side of Figure 50. The vertices $a$ and $b$ need not be valence two, either or both may have a valence greater than two.
Figure 50:Move 2.2 on a weighted planar tree.

From here we will show that canonization of minimal trees increases TCN complexity in a controlled manner.

Proof 2

Let $\Gamma$ be a minimal TCN arborescent representative of its equivalence class. Starting with the weight condition (W) maximally apply $F_3$ consolidating the weights of each vertex. $F_3$ may need to be reapplied after application of arithmetic moves and does not impact TCN.

Next we handle the stick condition starting with condition S.0 concerning sticks having non-zero weights. Maximally apply to $\Gamma$ moves 0.1 and 0.2; this removes zero weights of sticks. In the general canonization process S.A, the alternating stick condition, is handled by application of move 1.2. However, on a pair of vertices violating condition S.A, move 1.2 decreases the TCN. This means by minimality of $\Gamma$ , condition S.A must already be satisfied. Finally to obtain condition S.1, we apply moves 1.1 and 1.2, where move 1.2 may increase crossing number by 1.

The last condition to enforce is P, the positivity condition, which is done by modifying $\Gamma$ by application of moves 2.1 and 2.2. Similarly to our S.1 case, $\Gamma$ is minimal, 2.1, and 2.2 cannot decrease TCN. However, 2.1 and 2.2 may increase TCN by 1 and 2 respectively.

We note that Theorem 6 indicates that any opportunities to decrease TCN in a CWPTT are found on the end vertices of sticks adjacent to an essential vertex. Particularly, at least one weight used in the execution of 1.2, 2.1, and 2.2 must be carried by an essential vertex. Reversing the sequence of moves in Theorem 6 tells us that a minimal tree can be constructed from a CWPTT by application of TCN decreasing moves 1.2, 2.1, and 2.2. It is important to note that this does not guarantee that every set of applications of the 1.2, 2.1, and 2.2 moves to a CWPTT minimizes the TCN, only the existence of a path to a minimal tree.

Bounding complexity¶

We now introduce a bound on complexity between a CWPTT and a minimal representation of that tree. We build the bound by identifying the maximum number of subtrees of a CWPTT which admit a 1.2, 2.1, or 2.2 move. We begin by identifying the smallest TCN of a subtree that admits each move. These minimal subtrees follow directly from Definition~\ref{minimal-def-arithmetic} and our essential vertex requirement. The smallest, by TCN, canonical subtree admitting move 1.2 is 7, as in Figure 51a. For move 2.1 the smallest TCN for a canonical subtree is 5, as in Figure 51b, and for 2.2 the smallest TCN is 10, as in Figure 51c. Combining these smallest TCN subtrees with the how each move decreases TCN gives the bound in (11), where $\Gamma_m$ is a minimal representative for the tangle class of $\Gamma$ .

\begin{aligned} \text{TCN}\LP\Gamma\RP-\text{TCN}\LP\Gamma_m\RP&\leq 1\cdot\left\lfloor\frac{\text{TCN}\LP\Gamma\RP}{7}\right\rfloor+1\cdot\left\lfloor\frac{\text{TCN}\LP\Gamma\RP}{5}\right\rfloor+2\cdot\left\lfloor\frac{\text{TCN}\LP\Gamma\RP}{10}\right\rfloor\\ &\leq \frac{\text{TCN}\LP\Gamma\RP}{7}+\frac{\text{TCN}\LP\Gamma\RP}{5}+2\cdot\frac{\text{TCN}\LP\Gamma\RP}{10}\\ &=\frac{\text{TCN}\LP\Gamma\RP\cdot 19}{35}\\ \end{aligned}

(11)

6.2.3Right Leaning Identity CWPTT¶

The CWPTT are sufficient for distinguishing any two arborescent tangles via moves on their trees. Unfortunately, the equivalence class of CWPTT is still too large for computational enumeration to be feasible. The time required for pairwise comparisons grows badly exponentially. Luckily, from the class of CWPTT for an arborescent tangle, we can select a unique preferred form that allows for efficient direct enumeration by computer. To achieve this we will define two additional conditions for CWPTT, first the right leaning condition, and second, the identity condition. We call these preferred CWPTT Right Leaning Identity Canonical Weighted Planar Tangle Trees (RLITT).

6.2.3.1Existence of Right Leaning CWPTT¶

We start our construction of RLITT by defining what conditions make a CWPTT a right leaning CWPTT.

Our next step is to show that every arborescent tangle has a right leaning representative.

Proof 3

Let $\Gamma$ be a CWPTT representative for a tangle $T$ . If every weight $w_i$ of $\Gamma$ is in the highest indexed region of $\Gamma_{w_i}$ , we are done. Otherwise, we will follow a similar algorithm to that outlined by Bonahon and Siebenmann 8 for distinguishing CWPTT. Let $w_i$ be the weight for the lowest indexed vertex $v_i$ not in its highest indexed region of $\Gamma_{v_i}$ . With move $F_3^\prime$ shift $w_i$ so that it lies in the highest indexed region. Further, choose to shift $w_i$ anti-clockwise, this ensures that $v_j$ with $j<i$ are unchanged when $w_i$ is even. We repeat this process for any $v_k$ with $i<k$ where the weight $w_k$ is not in the highest indexed region. Since $\Gamma$ has finite vertices, the algorithm terminates with a $\Gamma$ transformed into a right leaning tree completing the proof.

6.2.3.2Existence of Identity CWPTT¶

Our second step in the construction of RLITT is to define what conditions make a CWPTT an identity CWPTT.

Again, we must show that every arborescent tangle has an identity representative.

Proof 4

Let $\Gamma$ be a CWPTT representative for a tangle $T$ . If the label $\alpha$ for the free bond of $\Gamma$ is $\iota$ we are done. Otherwise, we fall into one of three cases:

$\alpha=\zeta$ : In the case $\alpha$ is $\zeta$ we apply move $F_1$ . This modifies $\alpha$ by $\zeta$ yielding $\alpha\zeta=\zeta\zeta=\iota$ .
$\alpha=\eta$ : In the case $\alpha$ is $\eta$ we apply move $F_{2e}$ . This modifies $\alpha$ by $\eta$ yielding $\alpha\eta=\eta\eta=\iota$ .
$\alpha=\xi$ : In the case $\alpha$ is $\xi$ we apply move $F_{2o}$ . This modifies $\alpha$ by $\xi$ yielding $\alpha\xi=\xi\xi=\iota$ .

This transforms $\Gamma$ into an identity tree completing the proof.

6.2.3.3Existence of Right Leaning Identity CWPTT (RLITT)¶

What we have shown is that every arborescent tangle has at least one right leaning CWPTT and at least one identity CWPTT representative. Combining these two ideas, we will show that every arborescent tangle has at least one CWPTT that is right leaning and identity, we call such a CWPTT an RLITT.

6.2.3.4Uniqueness of Right Leaning Identity CWPTT¶

Our final step at identifying a preferred representative CWPTT of an arborescent tangle is to show that a $\LP +\RP$ -RLITT is unique in the set of CWPTT representing an arborescent tangle. We will utilize the key proposition from Bonahon and Siebenmann 8 a consequence of Theorem 3 and Theorem 4.

Proposition 1 (Bonahon and Siebenmann, Proposition 12.2 8)

Let $\Gamma$ and $\Gamma^\prime$ be $\LP +\RP$ -CWPTT tangle trees with isomorphic underlying abstract trees. Further, let $\varphi$ be a sequence of moves of the calculus of arborescent tangles ( $F_1$ , $F_2$ , $F_3^\prime$ , and the modified ring moves $\LP\pm R\RP$ ). Assume that there is an $i > 0$ such that $\varphi$ respects the cyclic orders of weight and bonds at each vertex $v_j$ with $j < i$ , and that the labels of the free bond $\alpha$ in $\Gamma$ and of $\varphi\LP \alpha\RP$ in $\Gamma^\prime$ are identical. Assume moreover, that one of:

$\varphi$ reverses the cyclic order of bonds at $v_i$
$\varphi$ does not respect the label in $V_4$ of some free bond of a vertex $v_j$ with $1 \leq j < i$ .

Then $\varphi$ is not a sequence of moves of the calculus of arborescent tangles taking $\Gamma^\prime$ to $\Gamma$ .

Proof 6

Let $\Gamma$ and $\Gamma^\prime$ be two $\LP +\RP$ -RLITT representatives for an arborescent tangle $T$ . Assume for the sake of contradiction that $\Gamma\neq \Gamma^\prime$ , meaning $T$ has two distinct $\LP +\RP$ -RLITT. The classification result in Theorem 3 and algorithm given by Theorem 4 we produce a sequence of moves in the calculus of arborescent tangles that takes $\Gamma^\prime$ to $\Gamma$ . Let $\varphi$ be such a sequence. By construction the labels in $V_4$ of $\Gamma$ and $\Gamma^\prime$ agree. Now, since $\Gamma\neq \Gamma^\prime$ there must be a first, in the total order, vertex $v_i$ where $\Gamma$ and $\Gamma^\prime$ disagree. As $\LP +\RP$ -RLITT, the location of weights for $v_i$ in $\Gamma$ and $\Gamma^\prime$ must appear in the same region. This requires that the disagreement at $v_i$ must be in cyclic order of its children. We find ourselves in the first case of Proposition 1, making $\varphi$ not a sequence of moves of the calculus of arborescent tangles taking $\Gamma^\prime$ to $\Gamma$ .

6.2.4Computational Methods¶

6.2.4.1An Encoding Strategy for Arborescent Knots and Tangles¶

The various flavors of weighted planar trees we have seen thus far are a useful tool for manipulation of arborescent tangles by humans or machines. Unfortunately, the tree structure is quite difficult to store directly in a computer database. We will rectify this by introducing a linearization strategy for weighted planar trees. This linearization strategy is designed to encode not only CWPTT but arbitrary weighted planar tangle trees. If a weighted planar tangle tree has more than one free bond we list the label as a subtree. We will omit this from our algorithm description as we are primarily concerned with RLITT.

We will descend the tree following the indexing of the total order (Indexing the total order of a tree). As we descend the tree, we annotate the linearization with two sets of delimiters. Each delimiter communicates extra information about the type of subtree it is delimiting, the two sets of delimiters are as follows:

$\LB\ \RB$ : Corresponds to a half-open proper stick and is interpreted as a twist vector for a rational tangle Kauffman & Lambropoulou (2002).
$\LP\ \RP$ : Corresponds to a vertex not on a half-open stick.

We will now walk through an example of the linearization algorithm. Let $\Gamma$ be a weighted planar tangle tree seen in Figure 52. As we walk the tree, the vertex currently being linearized will be called the object vertex.

Tangle Linearization Example¶

We begin by adding the $V_4$ label for our tangle to the linearization. We then start the following algorithm with the root as the object vertex.

For the object vertex, we add a ‘ $\LP\ \right.$ ’ delimiter to our linearization. Adding to our linearization the weights and children of the object vertex in an anti-clockwise order. When a child bond is encountered, we descend to that child. When we descend, we have two cases to consider, the child is a half-open proper stick or otherwise.

Case 1: The Child Is The Root Of A Half-Open And Proper Stick¶

When the child is the root of a is proper and half-open (contains a leaf vertex), we append that stick as the twist vector (Definition 2) for the corresponding rational tangle. Let the stick consist of the vertices, $v_i\cdots v_{i+k}$ , and weights, $w_i\cdots w_{i+k}$ . We delimit the stick with $\LB\ \RB$ , with each weight separated by a space, and the leaf weight as the left most entry of the twist vector. Further, every other entry is multiplied by -1, forcing the sign of all entries to match, as in (12).

\LB w_{i+k}\ \m w_{i+k-1}\cdots\ \m w_{i-1}\ w_{i}\RB

(12)

Case 2: The Child Is Essential, On A Open Stick, Or On A Non-Proper Stick¶

In this case, we restart the algorithm from the beginning with the current vertex as the object vertex.

When we have exhausted the children for the object vertex, we close our linearization for that vertex with the delimiter ‘ $\LN\ \RP$ ’. We then return to the parent linearization, repeating until all vertices have been exhausted. An example of a tree encoded with this strategy can be seen in Figure 52.

Figure 52:Encoded tree subtrees are indicated by color.

6.2.4.2Generation of Right Leaning Identity Weighted Planar Tangle Trees¶

Generation of Rooted Plane Tree¶

Just as rooted plane trees serve as the scaffolding we built WPTT on, a rooted plane tree algorithm will serve as the backbone for our RLITT generation algorithm. We will now give a brief description of the generation algorithm for rooted plane trees given by Nakano 14. We begin by defining the rightmost path of a rooted plane tree $\Gamma$ .

Next, we define a grafting operation on rooted plane trees $\Gamma$ and $\Gamma^\prime$ .

Definition 31

Let $\mathcal{T}^{p}_{n}$ be the set of rooted plane trees on $n$ vertices, $\Gamma_r\in \mathcal{T}^{p}_{n}$ , and $\Gamma_s\in \mathcal{T}^{p}_{m}$ . Define the grafting operation as follows.

\begin{aligned} \star_i:\mathcal{T}^{rp}_{n}\times\mathcal{T}^{rp}_{m}&\to\mathcal{T}^{rp}_{n+m}\\ \Gamma_r\times\Gamma_s&\mapsto\Gamma_r\star_i\Gamma_s \end{aligned}

At the vertex $v_i$ of $\Gamma_r$ , introduce an edge to the root of $\Gamma_s$ . Now, adjust the indexing of a vertex $s_k$ of $\Gamma_s$ as $v_{n+k}$ , placing $\Gamma_s$ as the rightmost child of $v_i$ . This results in a rooted plane tree $\Gamma\in \mathcal{T}^{rp}_{n+m}$ .

When grafting at the root $v_0$ we omit the index label in the grafting operation, that is, $\star_0$ is written simply as $\star$ . We call $\Gamma_r$ the rootstock and $\Gamma_s$ the scion.

Now we are prepared to give the algorithm to generate all rooted plane trees of a given size that are created from $\Gamma$ by a sequence of $\star_\ast$ operations on the rightmost path of $\Gamma$ .

This algorithm can be extended to an algorithm that finds all rooted plane trees up to a given size as follows.

Modification for RLITT¶

The algorithm described above serves as the inspiration for the algorithm we will build now for the enumeration of the arborescent tangles. Building this algorithm begins with modifying the grafting $\star_i$ operation to operate on weighted planar tangle trees as follows.

A rootstock \Gamma_r=\iota\LP \LP2\LP2\LB3\RB2\LB -3\RB3\RP \LB3\RB4\RP 4\RP in grey and scion \Gamma_s=\iota\LB 10\ 9 \RB in orange. Each vertex is
labeled with its index in the order on \Gamma. — (a)A rootstock $\Gamma_r=\iota\LP \LP2\LP2\LB3\RB2\LB -3\RB3\RP \LB3\RB4\RP 4\RP$ in grey and scion $\Gamma_s=\iota\LB 10\ 9 \RB$ in orange. Each vertex is labeled with its index in the order on $\Gamma$ .

Grafting at v_2 yields \Gamma_r\star_2\Gamma_s=\iota\LP \LP2\LP2\LB3\RB2\LB -3\RB\LB 10\ 9 \RB 3\RP \LB3\RB4\RP 4\RP — (a)A rootstock $\Gamma_r=\iota\LP \LP2\LP2\LB3\RB2\LB -3\RB3\RP \LB3\RB4\RP 4\RP$ in grey and scion $\Gamma_s=\iota\LB 10\ 9 \RB$ in orange. Each vertex is labeled with its index in the order on $\Gamma$ .

Just as we have adjusted the grafting operator, we must adjust the Nakano algorithm so it is aware of the extra data in an RLITT. The initial reaction to this problem may be to simply annotate the scion of the grafting operation with the weights necessary to reach a target TCN. Unfortunately, this method quickly runs into issues generating even just the Montesinos tangles, a smaller class of tangle described by 8 We must make a slightly more radical change to the Nakano algorithm, that being, grafting an entire RLITT to only the root $v_0$ of the rootstock. We will now show that a list of integral tangles (single vertex RLITT) Section 5.3.3 combined with grafting at the root generates all $\LP+\RP$ -RLITT. To start, we will prove that each $\LP+\RP$ -RLITT is integral or the result of grafting weighted planar tangle trees at the root.

Proof 7

Let $\Gamma$ be a $\LP +\RP$ -RLITT, we have three cases based on the valence of $v_0\in \Gamma$ .

The valence of $v_0$ is:

One: $\Gamma$ is integral (Integral Tangles), we fall into the first condition.
Two: $\Gamma$ is a stick at the root, we let $\Gamma_r=\iota\LB w_0\RB$ and $\Gamma_s=\iota\LP\alpha w_1\RP$ , where $\alpha$ is the remaining vertices, weights, and bonds of $\Gamma$ . $\Gamma_r$ is integral, and as such is $\LP+\RP$ -RLITT. Now to show that $\Gamma_s$ is $\LP+\RP$ -RLITT. Since $\Gamma$ is $\LP+\RP$ -RLITT, each $v_i$ in $\Gamma$ is canonical. Each vertex in $\Gamma_s$ is also in $\Gamma$ , so $\Gamma_s$ has only canonical vertices and by Theorem 5 is $\LP+\RP$ -canonical. Finally, since locations of weights are unchanged and grafting requires the scion to have identity label $\Gamma_s$ is $\LP+\RP$ -RLITT.
Three: The root of $\Gamma$ has two children, we let $\Gamma_s$ be the tree with the right child of $v_0$ as a root and $\Gamma_r=\iota\LP \alpha w_0\RP$ , where $\alpha$ is the remaining vertices, weights, and bonds of $\Gamma$ . Showing $\Gamma_s$ is $\LP+\RP$ -RLITT follows identically as it did in the first form.
We now show $\Gamma_r$ is $\LP+\RP$ -RLITT. First, given that $\Gamma$ is $\LP+\RP$ -RLITT, each $v_i$ in $\Gamma$ is canonical, consequently each vertex that is unchanged in $\Gamma_r$ ( $\LS v_i\RS_{i=2}^n$ ) is canonical. The vertices that differ in $\Gamma_r$ are $v_0$ and $v_1$ , $v_0$ remains the root of $\Gamma_r$ so is canonical. Showing the canonicity of $v_1$ depends on the the valence of $v_1$ . When the valence is 0 or is greater than 2, $\Gamma_r$ is $\LP+\RP$ -RLITT by the same argument as $\Gamma_s$ . When $v_1$ is valence 1 or 2, $\Gamma_r$ begins with a stick, and $v_1$ continues to satisfy the weight, positivity, and the $\pm 1$ and 0 portion of the stick condition. For the sign condition, when the signs agree, we are in form 2.1 of this theorem, and when they disagree, we are in form 2.2.
Greater than Three: The final case is when the valence of $v_0$ is greater than three. We let $\Gamma_s$ be the tree with the right child of $v_0$ as a root and $\Gamma_r=\iota\LP \alpha w_0\RP$ , where $\alpha$ is the remaining vertices, weights, and bonds of $\Gamma$ . Both $\Gamma_r$ and $\Gamma_s$ follow the same argument used for $\Gamma_s$ of the valence 1 case.

An identical theorem can be phrased for the $\LP-\RP$ -RLITT case. With this result, we can start our construction of a grafting algorithm for arborescent tangles.

Executing the algorithm with $\Gamma_r=\LS \iota\LB 1\RB\RS$ as the rootstock and $\Gamma_s=\LS \iota\LB 2\ 0\RB\RS$ produces the tangle $\iota\LB 2\ 0\ 1\RB$ . Unfortunately, this resultant tangle violates the stick condition and hence is not canonical. The remainder of this subsection will refine the grafting algorithm to satisfy each of the RLITT conditions.

Weight Condition¶

The simplest condition to verify is the weight condition. By construction, grafting the rootstock and scion introduces no additional weights at the grafting vertex. Meaning, the weight condition is satisfied with no adjustment to the algorithm.

Identity Condition¶

The next RLITT condition to address is the identity condition. We note that the $\star_i$ operation does not modify the $V_4$ label of the rootstock. This observation means if the rootstock is identity, the grafted tree will also be identity.

Stick Condition¶

To start with the stick condition we will prove that if grafting produces a non-canonical tree the non-canonical vertex must be adjacent to the root.

Proof 8

We need to show that each vertex $v_i$ at a distance greater than one from the root of $\Gamma=\Gamma_r\star\Gamma_s$ is canonical. The vertex $v_i$ is also a vertex of either $\Gamma_r$ or $\Gamma_s$ . If the vertex is in $\Gamma_r$ , then $v_i$ has a parent in $\Gamma_r$ and if the valence of $v_i$ is 2 or more $v_i$ also has children in $\Gamma_r$ . The parent and children in $\Gamma$ are the same as the parent and children in $\Gamma_r$ . Since $\Gamma_r$ is RLITT $v_i$ is canonical in $\Gamma_r$ and since it shares a parent and children in $\Gamma$ it is also canonical in $\Gamma$ . Similarly, for $v_i$ in the scion, showing canonicity of grafting is dependent only on the canonicity of the vertices at a distance up to one from the root.

The $\LP-\RP$ -RLITT case is shown identically, covering the majority of possibilities. However, we need to take special care when grafting a non-negative to a non-positive tree (or the reverse). Before we address that case we define a restricted class of scions that, after grafting, satisfy the nonzero portion of the stick condition.

Positivity/Negativity Condition¶

Our approach to the positivity and negativity condition follows our approach to the stick condition. We will leverage Theorem 12 to add a check for positivity and negativity in our algorithm.

Algorithm 11 (Find weighted planar trees by grafting

\LP-\RP

-RLITT good scions to the root of

\LP-\RP

-RLITT rootstocks)

Input

A collection of $\LP-\RP$ -RLITT good scions $T_s$
A collection of $\LP-\RP$ -RLITT rootstocks $T_r$

Output

A collection of weighted planar trees (still not guaranteed to be RLITT)

Routine

For each combination of $\Gamma_s\in T_s$ and $\Gamma_r \in T_r$
1. Compute $\Gamma = \Gamma_r\star \Gamma_s$
2. For each vertex $v_i$ at distance 1 from the root of $\Gamma$
  1. Continue to the next iteration of the outer loop if $v_i$ fails to satisfy the stick condition
  2. Continue to the next iteration of the outer loop if $v_i$ fails to satisfy the negativity condition
3. Report $\Gamma$

Right Leaning Condition¶

Satisfying the right leaning condition is a consequence of the modified $\star_i$ operation. Our definition of the $\star_i$ operation grafts the scion in such a way that weights at $v_i$ are always right of the scion. To fully satisfy the right leaning condition, we need to ensure that any stick subtrees are in the right most positions. This is accomplished with a slight modification of our grafting algorithms.

Algorithm 12 (Find weighted planar trees by grafting

\LP+\RP

-RLITT good scions to the root of

\LP+\RP

-RLITT rootstocks)

Input

A collection of $\LP+\RP$ -RLITT good scions $T_s$
A collection of $\LP+\RP$ -RLITT rootstocks $T_r$

Output

A collection of RLITT

Routine

For each combination of $\Gamma_s\in T_s$ and $\Gamma_r \in T_r$
1. Compute $\Gamma = \Gamma_r\star \Gamma_s$
2. Shift ring subtrees of the root of $\Gamma$ to the right
3. For each vertex $v_i$ at distance 1 from the root of $\Gamma$
  1. Continue to the next iteration of the outer loop if $v_i$ fails to satisfy the stick condition
  2. Continue to the next iteration of the outer loop if $v_i$ fails to satisfy the positivity condition
4. Report $\Gamma$

Algorithm 13 (Find weighted planar trees by grafting

\LP-\RP

-RLITT good scions to the root of

\LP-\RP

-RLITT rootstocks)

Input

A collection of $\LP-\RP$ -RLITT good scions $T_s$
A collection of $\LP-\RP$ -RLITT rootstocks $T_r$

Output

A collection of RLITT

Routine

For each combination of $\Gamma_s\in T_s$ and $\Gamma_r \in T_r$
1. Compute $\Gamma = \Gamma_r\star \Gamma_s$
2. Shift ring subtrees of the root of $\Gamma$ to the right
3. For each vertex $v_i$ at distance 1 from the root of $\Gamma$
  1. Continue to the next iteration of the outer loop if $v_i$ fails to satisfy the stick condition
  2. Continue to the next iteration of the outer loop if $v_i$ fails to satisfy the negativity condition
4. Report $\Gamma$

Full Generation Algorithm¶

The algorithm we have developed so far generates new RLITT from two restricted collections of trees. Unfortunately, it doesn’t yet tell us how to select the collections that guarantee the generation of all arborescent tangles up to a target TCN are represented. The final step in the generation scheme is describing how to build these collections. With computer enumeration in mind, we would like for our strategy to be easily split into jobs that can be run in parallel.

We observe that when grafting $\Gamma_r\star_i\Gamma_s=\Gamma$ , the TCN $r$ of $\Gamma_r$ and the TCN $s$ of $\Gamma_s$ sum to the TCN of $\Gamma$ . This observation is the key underpinning of the strategy we use to define discrete generation jobs. For a target TCN, we have the integer pairs seen in (13) that sum to the target TCN. Each of these pairs defines two classes, determined by TCN, of RLITT that can be grafted.

\begin{aligned} (0&,\text{TCN})\\ ( 1&,\text{TCN}-1)\\ &\vdots\\ (\text{TCN}-1&,1)\\ (\text{TCN}&,0) \end{aligned}

(13)

The next question we should ask is if we can simplify the list at all. First, as we saw in the discussion of the stick condition Stick Condition, we need our scions to be good. This means we cannot have 1 or 0 in the second position of the pair, excluding $(\text{TCN}-1,1)$ and $(\text{TCN},0)$ from the list. Second, the pair $(0,\text{TCN})$ will be excluded from our list, since the zero-crossing tangle $\iota[0\ 0]$ can’t serve as rootstock; grafting any scion would violate the stick condition. We will recover tangles with root weight 0 with a post-processing step.

The following is the recursive algorithm used to take us from the set of RLITT with $\text{TCN}-1$ to the set of RLITT of the target TCN.

Algorithm 14 (Find RLITT of given TCN from all RLITT of TCN-1)

Input

A target TCN
All RILTT up to TCN-1

Output

A set $T$ of all RLITT of TCN

Routine

Set $i=1$
Set $N=\text{TCN}-1$
Set $T$ to the set $\LS \iota[\text{TCN}], \iota[0\ \text{TCN}]\RS$
For each pair $(i,N-i)$
1. Set $T_{s+}$ to be the set of $+$ -RLITT good scions with TCN equal to $\text{N}-i$
2. Set $T_{s-}$ to be the set of $(-)$ -RLITT good scions with TCN equal to $\text{N}-i$
3. Set $T_{r+}$ to be the set of $(+)$ -RLITT TCN equal to $i$
4. Set $T_{r-}$ to be the set of $(-)$ -RLITT TCN equal to $i$
5. Execute Algorithm 13 with input $T_{r-}$ and $T_{s-}$
6. Add the results to $T$
7. Execute Algorithm 12 with input $T_{r+}$ and $T_{s+}$
8. Add the results to $T$
For every RLITT $\Gamma$ in $T$
1. Continue to the next iteration of the loop if root of $\Gamma$ is valence two with weight zero
2. Compute $\iota[0]\star\Gamma$ and add to T

It is important to note that the output of Algorithm 15 includes duplicates in the form of $\LP+\RP$ -RLITT and $\LP-\RP$ -RLITT pairs. To deduplicate our list so it contains only topologically unique objects, we select from the list the collection of $\LP+\RP$ -RLITT.

Footnotes¶

Each tangle can be represented by an infinite number of diagrams. A unique representative in this context means a particular “flavor” of diagram that exists for every tangle in the class we are concerned about.
↩

References¶

Conway, J. H. (1970). An Enumeration of Knots and Links, and Some of Their Algebraic Properties. Computational Problems in Abstract Algebra, 329–358. 10.1016/B978-0-08-012975-4.50034-5
Goldman, J. R., & Kauffman, L. H. (1997). Rational Tangles. Advances in Applied Mathematics, 18(3), 300–332. 10.1006/aama.1996.0511
Rockett, A. M., & Szüsz, P. (1998). Continued Fractions (2. repr). World Scientific.
Burde, G., Zieschang, H., & Heusener, M. (2013). Knots. DE GRUYTER. 10.1515/9783110270785
Kauffman, L. H., & Lambropoulou, S. (2002). On the Classification of Rational Knots. arXiv: Geometric Topology. 10.48550/ARXIV.MATH/0212011
Bryhtan, Z. C. (2024). Tabulating 2-String Tangles with Methods to Count and Generate Them. 10.25820/ETD.007313
Ernst, C. (1996). TANGLE EQUATIONS. Journal of Knot Theory and Its Ramifications, 05(02), 145–159. 10.1142/S0218216596000114
Bonahon, F., & Siebenmann, L. C. (2016). New Geometric Splittings of Classical Knots and the Classiﬁcation and Symmetries of Arborescent Knots. https://dornsife.usc.edu/francis-bonahon/wp-content/uploads/sites/205/2023/06/BonSieb-compressed.pdf
Connolly, N. (2021). Classification and Tabulation of 2-String Tangles: The Astronomy of Subtangle Decompositions. University of Iowa. 10.17077/etd.005978
Caudron, A. (1982). Classification Des Noeuds et Des Enlacements. Université de Paris-Sud, Dép. de mathématique. https://books.google.com/books?id=W1nvAAAAMAAJ
Gren, B. A., Sulkowska, J. I., & Gabrovšek, B. (2025). Classification of Algebraic Tangles. https://arxiv.org/abs/2504.06901
Łukasiewicz, J. (1929). Elementy Logiki Matematycznej. Nakładem komisji wydawniczej Koła matematyczno-fizycznego słuchaczów Uniwersytetu warszawskiego.
Tait, P. G. (1900). On Knots I, II, III. Scientific Papers London: Cambridge University Press, 1, 273–347.
Nakano, S. (2002). Efficient Generation of Plane Trees. Information Processing Letters, 84(3), 167–172. 10.1016/S0020-0190(02)00240-5

6 Tabulation

6.1The First Tangle Datasets¶

6.1.1Rational Tangles¶

6.1.1.1Construction¶

Correspondence With Extended Rational Numbers¶

6.1.1.2Canonical Twist Vectors¶

6.1.1.3Computational Methods¶

Notation¶

Generation¶

6.1.2Montesinos Tangles¶

6.1.2.1Construction¶

Unique Representative¶

6.1.2.2Computational Methods¶

Notation¶

Generation¶

6.2Arborescent Tangles¶

6.2.1Definition of Arborescent¶

6.2.2Weighted Planar Trees¶

6.2.2.1Construction of Arborescent Knots from Weighted Planar Trees¶

Bands and Plumbing Squares¶

Plumbing bands¶

Weighted Planar Trees¶

Indexing the total order of a tree¶

Weighted Planar Tangle Trees¶

6.2.2.2Anatomy of a tree¶

Ring subtree¶

Essential Vertex¶

Sticks of a Tree¶

Integral Tangles¶

Rational Tangles¶

Tree Crossing Number¶

6.2.2.3Calculus of Weighted Planar Trees¶

The F3′F^\prime_3F3′​ Move¶

F3′F_3^\primeF3′​ on bands¶

F3′F_3^\primeF3′​ Examples¶

The F2F_2F2​ Move¶

The F1F_1F1​ Move¶

The R±R^\pmR± moves¶

6.2.2.4Canonical Weighted Planar Tangle Trees (CWPTT)¶

Canonical Vertex¶

6.2.2.5Minimalization of CWPTT¶

CWPTT are Not Minimal¶

Canonization Can Increase Complexity¶

Bounding complexity¶

6.2.3Right Leaning Identity CWPTT¶

6.2.3.1Existence of Right Leaning CWPTT¶

6.2.3.2Existence of Identity CWPTT¶

6.2.3.3Existence of Right Leaning Identity CWPTT (RLITT)¶

6.2.3.4Uniqueness of Right Leaning Identity CWPTT¶

6.2.4Computational Methods¶

6.2.4.1An Encoding Strategy for Arborescent Knots and Tangles¶

Tangle Linearization Example¶

Case 1: The Child Is The Root Of A Half-Open And Proper Stick¶

Case 2: The Child Is Essential, On A Open Stick, Or On A Non-Proper Stick¶

6.2.4.2Generation of Right Leaning Identity Weighted Planar Tangle Trees¶

Generation of Rooted Plane Tree¶

Modification for RLITT¶

Weight Condition¶

Identity Condition¶

Stick Condition¶

Positivity/Negativity Condition¶

Right Leaning Condition¶

Full Generation Algorithm¶

The $F^\prime_3$ Move¶

$F_3^\prime$ on bands¶

$F_3^\prime$ Examples¶

The $F_2$ Move¶

The $F_1$ Move¶

The $R^\pm$ moves¶