University Seminar: Logic Across Disciplines- Multiplying Fractions in a Topological Way

Title:  Multiplying Fractions in a Topological Way
Speaker:   Jozef Przytycki, GWU -
Date and Time:  Friday, September 21 2018, 01:00pm-02:00pm
Place: Rome Hall (801 22nd Street), Room 771

Abstract:  I describe the work of our Mathathon group (Sujoy Mukherjee, Rhea Palak Bakshi, Marithania Silvero, Xiao Wang), December 2017-- April 2018 on an algebra structure (Kauffman bracket skein algebra) of links in thickened 4-holed sphere.

Based on the presentation of the Kauffman bracket skein module of the torus which I gave in 1987 when the theory of skein modules was just invented, Charles D. Frohman and Razvan Gelca established a complete description of the multiplicative operation leading to a famous product-to-sum statement: "the KBSA of a torus is a quantum torus". In this talk, 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 (motivated by E.Witten work).
The talk will be elementary and all needed notions defined (compare e-print: \ {\tt arXiv:1805.06062 [math.GT]}).