On multiplying curves in the Kauffman bracket skein algebra of the thickened four-holed sphere

TL;DR

This paper investigates the multiplicative structure of the Kauffman bracket skein algebra of the thickened four-holed sphere, providing an algorithm for multiplication, explicit formulas for certain curves, and conjectures on basis positivity.

Contribution

It introduces an algorithm for multiplying elements in the skein algebra of the four-holed sphere and offers explicit formulas for specific curve families, advancing understanding of its algebraic structure.

Findings

Algorithm for computing products with quasi-polynomial growth

Explicit formulas for families of curves

Conjecture on the existence of a positive basis

Abstract

Based on the presentation of the Kauffman bracket skein module of the torus given by the third author in previous work, Charles D. Frohman and R\u{a}zvan Gelca established a complete description of the multiplicative operation leading to a famous product-to-sum formula. In this paper, we study the multiplicative structure of the Kauffman bracket skein algebra of the thickened four-holed sphere. We present an algorithm to compute the product of any two elements of the algebra, and give an explicit formula for some families of curves. We surmise that the algorithm has quasi-polynomial growth with respect to the number of crossings of a pair of curves. Further, we conjecture the existence of a positive basis for the algebra.