Classification and tabulation of 2-string tangles
Nicholas Connolly, Isabel K. Darcy, Keiko Kawamuro, Benjamin Cooper, Charles Frohman, Ryan Kinser · 2021
Tangles are a type of knot theoretic structure originally introduced by Conway to tabulate knots. Intuitively, tangles can be thought of as the building blocks of mathematical knots. The two principal goals of this dissertation are classification and tabulation of 2-string tangle diagrams, and we approach both endeavors through the careful study of diagram notations. Broadly speaking, tangles can be divided into two types: algebraic and non-algebraic. Many special cases of algebraic tangle are well understood and have been fully classified. We begin by reviewing the structure of these special types of tangles, each of which may be uniquely represented using a canonical diagram. Unlike these special cases, generic algebraic and non-algebraic tangle diagrams are not fully classified. To better understand these types of tangles, we propose a graph-based notation for tangle diagrams, referred to as subtangle planar diagram (SPD) code. SPD code can be used to represent generic tangle diagrams using decompositions into subtangles. Based on this description, we show how algebraic and non-algebraic tangles can be distinguished using planar graphs. This description enables any algebraic diagram to be converted into an equivalent semi-canonical form. Furthermore, we show how certain types of planar graphs known as constellations arise in connection with subtangle decompositions for non-algebraic diagrams. All 2-string tangle diagrams with at most 9 crossings can be represented using a combination of tangle algebra and the first ten constellations. In addition to this theoretical work, we also discuss a number of related computational results. In particular, we give an overview of an algorithm to compute several variations of the SPD code, which has been successfully implemented as a program in C/C++. This algorithm has been used in combination with tangle tabulation programs to classify tangle diagrams. We conclude by discussing the progress on the prototype of a web-accessible database of 2-string tangles currently being developed to catalog these results.