# University Seminar- Computability, Complexity, and Algebraic Structure

Time: Wednesday, September 20, 11:15am – 12:15pm

Place: Monroe Hall, Room 250

Speaker: Jozef Przytycki, GWU

Title: * *The story of Fox colorings of link diagrams: the Kauffman-Harary Conjecture and beyond

Abstract: For a reduced alternating diagram of a knot with a prime determinant p, the Kauffman-Harary conjecture states that every non-trivial Fox p-coloring of the knot assigns different colors to its arcs. In this talk, we outline a history of our work on a generalization of the conjecture stated over twenty years ago by Asaeda, Przytycki, and Sikora: for every pair of distinct arcs in the reduced alternating diagram of a prime link with determinant $d$, there exists a Fox $d$-coloring that distinguishes them (Mattman and Solis solved the original conjecture in 2009). We sketch the solution of the generalized Kauffman-Harary conjecture by Mathathoners in December 2022.This was a joint work with Mathathoners VII: Rhea Palak Bakshi, Huizheng Guo, Gabriel Montoya-Vega, and Sujoy Mukherjee. Finally we analyze the space of Fox colorings and the Alexander-Burau-Fox module of the closure of the 3-braid (s1s2^{-1})^n and express the result using Fibonacci and Chebyshev polynomials, respectively.

This is a joint work with Ali Guo and Anthony Christiana.