The tabulation, creation of a list, of mathematical knots has been a core area of mathematics research for over 150 years. A key breakthrough in knot tabulation was Conway’s development of a building block for knots, the two string tangle. A two string tangle is a portion of a mathematical knot bound within a ball and intersecting that ball in exactly four points. A table of these objects has been called:

The most important missing infrastructure project in knot theory

• Dr. Dror Bar-Natan 1

Without such a table of tangles, mathematicians and scientists are in the position of a chemist who possesses a table of fatty acids but no periodic table. In this thesis we answer, in part, this call.

As we progress through this thesis, we produce successively larger and more complete tables of tangles. Each table we produce requires the development of the mathematical theory demonstrating that what we have produced is what we say it is. Then, for each tangle type we develop the theory for, we also develop a collection of software designs used to convince a computer to quickly generate large tables of tangles. We will also discuss ways that the accessibility of knot theory and in particular tabulation make this domain a candidate for undergraduate research. As part of our discussion of undergraduate research, we will outline a software engineering process particularly suited for undergraduate research in knot theory, as well as a model life cycle for an undergraduate research experience in computational knot theory.