Logic/Topology Seminar-Complexity of the Jones polynomial and Khovanov homology

Time: Wednesday, September 22, 4:45-6:00PM

Place: Bell Hall 204

Speaker:  Jozef Przytycki, GWU

Title: Complexity of the Jones polynomial and Khovanov homology

Abstract: The first GW Logic-Topology seminar was given by Zbyszek Oziewicz in February 2006 (the topic “Braided Logic”). Oziewicz died of Covid in December 2020. This talk is dedicated to Zbyszek.

It is well understood, starting from Francois Jeager work, that computing the Jones polynomial (hence also Khovanov homology) is NP-hard. For a braid of fixed number (say n) of strings all classical quantum invariants can be computed in polynomial time. We conjecture here a similar result for Khovanov homology.
Conjecture: Computational complexity of computing Khovanov homology for (closed) braids of a fixed number of strings, has polynomial time complexity with respect to number of crossings.
We describe a preliminary work on the conjecture, in particular we prove it for 3- and 4-braids and extreme Khovanov homology. We also prove the Wedge of Spheres conjecture for 4-braid diagrams.

We offer a gentle introduction to the topic including the definition of Simplicial Complex, its homology and geometric realization.
This is joint project with Marithania Silvero of Seville.