University Seminar: Logic Across Disciplines-Mathathon II: Search for Interesting Torsion in Khovanov Homology

Title:   Mathathon II: Search for Interesting Torsion in Khovanov Homology

Speaker:    Jozef Przytycki, GWU

http://home.gwu.edu/~przytyck/
Date and Time: Thursday, October 5, 2017, 2:30-3:30pm

Place: Rome Hall (801 22^nd Street), Room 771

Abstract:  We will describe the work of our Mathathon group (Sujoy Mukherjee, Marithania Silvero, Xiao Wang, Seung Yeop Yang), Dec. 2016–Jan. 2017, on torsion in Khovanov homology different from Z_2. Khovanov homology, one of the most important constructions at the end of XX century, has been computed for many links. However, computation is NP-hard and we are limited to generic knots of up to 35 crossings with only some families with larger number of crossings. The experimental data suggest that there is abundance of Z_2-torsion but other torsion seems to be rather rear phenomenon. The first Z_4 torsion appears in 15 crossing torus knot T(4,5), and the first Z_3 and Z_5 torsion in the torus knot T(5,6). Generally, calculations by Bar-Nathan, Shumakovitch, and Lewark suggest Z_p^k torsion in the torus knot T(p^k,p^k+1), p^k >3, but this has not yet been proven. We show, with Mathathoners, the existence of Z_n-torsion, n>3, for some infinite family of knots.  The simplest of them is obtained by deforming the torus knot T(5,7) by a t_2k-moves. We also prove the existence of knots with other torsion, the largest being Z_2^23, so the cyclic group of over 8 millions elements. We combine computer calculations (and struggle with NP hardness) with homological algebra technique.

The talk will be elementary and all needed notions will be defined.