University of California, San Diego
Subject Listing - Mathematics
Advisor: Dr. Justin Robert
Thursday, Poster Session 2, Presentation Kiosk 29 B, Health & Fitness Center
EXPLORATION OF KHOVANOV HOMOLOGY AND LINK COBORDISMS
Since the introduction of the Jones polynomials in knot theory, there has been a flurry of generalizations and other polynomials that are invariants of knots. Invariants help us classify and distinguish different knots. This research explores Khovanov's seminal work of a stronger invariant than the Jones polynomial. In his paper, Khovanov introduces what is typically referred to as Khovanov homology, and he showed how this is an invariant of links. Khovanov conjectured how an induced map on the homology groups of the target and source link of a link cobordism (4dimensional knotted surfaces), was invariant under isotopy UP TO A SIGN. Jacobsson proved Khovanov's conjecture and provided a "sign table" on Movie Moves (Reimdermeister type moves for four dimensional knotted surfaces). The research I am involved with deals with the exploration of Khovanov homology and link cobordisms to find a solution to this "Sign table" problem. Possible solutions we are still testing are: 1) To uncover a structure, that when added to the Movies Moves, will make the signs go away and, 2) Find a new definition of Khovanov's invariant for Framed link cobordisms that will make the signs consistent. Having taken care of the "Sign table," then we would have a trouble-free invariant that will allow us to calculate and retrieve more information from link cobordisms.
Advisor: Dr. Justin Robert, Associate Professor, Department of Mathematics, University of California, San Diego, La Jolla, CA