The enumeration of provably unique mathematical knots, tabulation, has been a core area of research for over 150 years. One important advancement in this area was Conway’s development of a building block for knots, the tangles. Formally, a n-string tangle is a portion of a knot diagram enclosed in a Jordan curve that intersects the knot in exactly 2n points. The tabulation of the two string tangles, Conway’s building blocks for knots, has been called:

"The most important missing infrastructure project in knot theory" - Dr. Dror Bar-Natan

Without such a table of tangles, researchers 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.

This thesis develops the theory needed to tabulate tables of successively more complex classes of tangles. We start with the rational and Montesinos tangles and then conclude with the arborescent tangles, often called the algebraic tangles. For each of these classes of tangles, we additionally develop a collection of software designs used to efficiently and scalably compute tables of these tangles to high crossing number (19). We will also discuss ways that the accessibility of knot theory and in particular tabulation make the 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.

This material is based upon work supported by the National Science Foundation under Award No. DMS-2038103.

Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.