Invariants for knots, and graphs on surfaces

Louisiana State University, Baton Rouge LA

Investigators

Abstract

In recent years the study of polynomial knot invariants like the Jones polynomial gained new momentum. In particular the Volume conjecture that claims a deep relationship between the Jones polynomial of cablings of the knot on one side and the hyperbolic volume of the knot complement on the other side led to a new point of view. The topics of the project are inspired by the Volume conjecture. The scope is to gain a better understanding of both the colored Jones polynomial and the hyperbolic volume. In earlier works of the principal investigator and his collaborators it was shown that bounds for the hyperbolic volume of certain classes of knots can be read off from coefficients of the colored Jones polynomial. This made it interesting to study the leading and trailing coefficients of the colored Jones polynomial. Under certain conditions on the knot there seems to be an infinite polynomial, depending on the knot, whose first n coefficients agree with the first n coefficients of the colored Jones polynomial at color n of that knot. The nature of these infinite polynomials as well as their number theoretical properties will be studied. Part of the project will also be to find a better topological understanding of the colored Jones polynomial. For this, earlier work of the principle investigator and his collaborators will be used that interprets the regular Jones polynomial as a state sum over subgraphs of a graph, embedded on an oriented surface, that is assigned to each knot diagram. Thus, every state is equipped with three parameters, the third being the genus of the subgraph. It has a long and fruitful tradition to study objects that are embedded in three dimensional space, e.g. knots, via their projections on a plane. However, information about the original object is lost and additional information is needed to indicate which arc of the knot is farther away from the projection plane. By projecting on other surfaces more information about the original object can be preserved. These projections will be used to gain understanding of the topological and geometrical properties of knot invariants like the Jones polynomial.