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.
Knot theory is crucial for the understanding of how 3-dimensional spaces can look like. Knots are embeddings of a circle in a space. Drilling out the neighborhood of a knot and gluing in a solid torus, i.e. a closed neighborhood of a circle, yields in general a different space. By iterating this process one can obtain all possible 3-dimensional spaces. A way to study knots is via their projections on a plane. However, the disadvantage is that a knot in general cannot be uniquely recovered from such a projection. At a self-intersection in the projection it is not clear which of the strands of the knot is farther away from the plane. Part of the project was to study more general projections of knots onto surfaces, e.g. onto a torus, such that the knot is essentially uniquely recoverable from the projection. This defines a complexity for knots: The simplest surface on which the knot can be projected such that it is essentially recoverable from the projection. The complexity was compared and related to other complexities and invariants in knot theory. For the second part of the project infinite sequences of integers were studied that one can assign to certain classes of knots. For many knots one can give an explicit description for these infinite sequences. An operation on knots was found that assigned to each pair of given knots a third knot. It was shown that the infinite sequence assigned to the third knot can be readily computed from the infinite sequences assigned to the two original knots. The project drew ideas from mathematical physics and quantum topology and yielded applications in number theory and combinatorics.
