Algorithms for skein manipulation and automation of skein computations

Abstract

Skein manipulations prove to be computationally intensive due to the exponential nature of skein relations. Resolving each crossing in a knot diagram produces 2 new knot diagrams; knot diagrams with over 5 crossings become increasingly difficult to work with. In this work, I introduce a method for automating these computations using algorithms developed to perform computations in the knot complement. This method is developed for all 2-bridge knots, particularly twist knots and (2,2p+1)-torus knots, but can be extended to other families with modification. After showing these algorithms produce the desired result, I demonstrate their implementation in a Python program. This program is used to to compute several known examples, demonstrating how results obtained through several months of work can be can now be obtained in less than 5 minutes. This program will be used for testing various hypotheses in $SU(2)$ Chern-Simons theory.

Embargo status: Restricted to TTU community only. To view, login with your eRaider (top right). Others may request the author grant access exception by clicking on the PDF link to the left.