Invariants for knots, and graphs on surfaces

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The Volume Conjecture claims a deep relationship between the Jones polynomial of cablings of a knot on one side and the hyperbolic volume of the knot complement on the other side. This project is inspired by the Volume Conjecture. The scope is to gain a better understanding of the colored Jones polynomial, and its relations to the geometry of the knot complement. 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 is a power series assigned to the knot that determines the first k coefficients of its colored Jones polynomial, for every fixed k and sufficiently large color. The investigator will study the geometric and number theoretic properties of these power series. Moreover, in earlier work of the investigator and his collaborators the Jones polynomial was interpreted as a state sum over subgraphs of a graph, embedded on an oriented surface, that is assigned to each knot diagram. The relation of the genera of those graphs and their subgraphs to properties of the colored Jones polynomial will be studied.When studying objects that are embedded in three dimensional space, e.g. knots, via their projections on a plane 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.