University Seminar- Computability, Complexity, and Algebraic Structure

Time: Wednesday, September 4, 2:20 – 3:20pm

Place: Rome Hall, Room 352

Speaker:  Jozef Przytycki, GWU

Title:  Montesinos-Nakanishi Conjecture for links up to 20 crossings

Abstract:  In 1981, Yasutaka Nakanishi, then a graduate student at Kobe University, formulated the following intriguing-sounding conjecture:

Every link is a 3-move equivalent to a trivial link.

The conjecture was proved in many special cases (e.g., Qi Chen proved it for 5-string braids with one exception of 20 crossings) but it was an open problem for over 20 years. In 2002, it was shown by Mieczyslaw Dabkowski and me that it does not hold in general and that the Chen link is a counterexample (the novel concept of the Burnside group of a link was introduced then). Recently, we proved the Montesinos Nakanishi conjecture for links up to 19 crossings and showed that it holds for links of 20 crossings except for Chen’s example and its mirror image. The tools we use are Conway’s basic polyhedra (classified up to 20 crossings), Vogel-Traczyk algorithm “from diagrams to braids,” analysis of conjugacy classes of 5 braids, Burnside groups of links, and, maybe most important, computer generated analysis of diagrams up to 20 crossings.  

This is a joint work with Rhea Palak Bakshi, Benjamin A. Burton, Huizheng (Ali) Guo, Dionne Ibarra, Gabriel Montoya-Vega, and Sujoy Mukherjee, which started as a Mathathon type project at MATRIX Institute in Creswick, Australia in June 2024.