Student seminar: The Jones polynomial, with a view to Khovanov homology

by Liam Watson (UCLA)

2013.03.11 | Jane Jamshidi

Date Wed 13 Mar
Time 16:15 17:00
Location Aud. D3 (1531-215)

Abstract:

An interesting problem in low-dimensional topology is to decide if a knot is truly knotted. There are a range of invariants one can apply in attempting to answer  questions pertaining to knottedness, and these tend to generate interesting algebraic structures. One example of such an invariant is the Jones polynomial, introduced in the 80s. Though this invariant is quite good at distinguishing knots, it is an outstanding problem to decide if a there are non-trivial knots with trivial Jones polynomial. More recently, Khovanov has provided a generalization of the Jones polynomial in the form of a homology theory; the Euler characteristic of this theory (suitable defined) recovers the Jones polynomial. Perhaps surprisingly, a recent result of Kronheimer and Mrowka establishes that this generalization does detect the trivial knot. This talk will be introductory, focusing on the definition of the Jones polynomial with a view to constructing Khovanov's generalization. 

 

IT, computer science and mathematics , Public/media, Staff, Students