Algorithms for Skein Manipulation in a Genus-2 Handlebody

Abstract

We present a series of algorithms for skein manipulation in a genus-2 handlebody, implementing a novel strand sorting method to reduce any skein to a skein in a 2-punctured disk. This reduction guarantees resolution as a linear combination of basis elements of the Kauffman Bracket Skein Module. Manually, these skein manipulations prove to be computationally intensive due to the inherent exponential nature of skein relations (i.e., a skein diagram with $n$ crossings yields $2^n$ new skein diagrams, each in $\mathbb{C}[t,t^{-1}]$, the Laurent polynomials with complex coefficients). Thus, as the number of crossings in a skein diagram increases, manual computations become intractable and automation desirable. We enable the automation of all skein computations in the genus-2 handlebody by first converting the skein diagram into an equivalent array, reducing the task of performing skein computations to that of implementing array operators, and then proving that we can always recover the resulting complex Laurent polynomial.