4:00 pm Tuesday, November 17, 2009
THESIS DEFENSE : Constructing Thick knots in Khovanov Homology
by Andrew Elliott (Rice Math Dept.) in KH 101A famous knot invariant of the mid 80's is the Jones polynomial. This invariant was originally conceived as the trace of a representation on the braid group, but since has been given several combinatorial descriptions, as well as interpretations in statistical mechanics via the Yang-Baxter equations. The Jones polynomial has been very successful in distinguishing knots; it classifies knots of 9 crossings or less, and easily distinguishes mirrors of knots, a property difficult for prior invariants to detect. My research deals with a generalization of the Jones polynomial called Khovanov homology. Khovanov renewed interest in "categorification" with a preprint circulated in 1998. At that time, he proposed a generalization of the Jones polynomial to a bigraded homology theory, in such a way that the ranks of the homology generalized the coefficients of the Jones polynomial. This categorification procedure has since been applied to many other polynomials both in topology and in combinatorics, from the Alexander polynomial of knot theory to the chromatic polynomial of graph theory. In each case, the new invariant constructed is stronger than the classical invariant, but even in the case of Khovanov homology, it remains an active area of research to examine just how much new information the categorification contains. One measuring stick for how much new data Khovanov homology carries over the Jones polynomial is called homological width. This counts the number of slope 2 diagonals on which the bigraded homology is supported. When the homology of a knot is supported on exactly two diagonals, the knot is called H-thin, and its reduced Khovanov homology is determined by its Jones polynomial and quantum intercept. From this perspective, the knots whose homology is supported on 3 or more diagonals are the ones for which Khovanov homology is an "interesting" invariant. Such knots are called H-thick, and understanding their structure is an active area of research. My thesis involves a new approach to the study of homological width and H-thick knots. Instead of attacking the problem with the long exact sequence for Khovanov homology, I study Khovanov homology classes which have state cycle representatives. By producing such nontrivial homology classes in different diagonals, I can get a lower bound on homological width for knots whose homology is too complex for computer calculation. As an application, I describe a general procedure, quasipositive modification, for constructing H-thick knots in rational Khovanov homology. Moreover, I show that specific families of such knots cannot be detected by previous methods, like Khovanov's thickness criteria. I also exhibit a sequence of prime links related by quasipositive modification whose homological width is increasing. Host Department: Rice University-MathematicsSubmitted by cochran@rice.edu