Invariants for knots, and graphs on surfaces

近年来，琼斯多项式等多项式纽结不变量的研究获得了新的动力。特别是卷猜想，声称琼斯多项式cablings的结的一方和双曲卷的结补充的另一方之间的深刻关系导致了一个新的观点。该项目的主题受到体积猜想的启发。范围是为了更好地理解有色琼斯多项式和双曲体积。在早期的作品的主要研究者和他的合作者，它表明，边界的双曲体积的某些类别的结可以读出的系数的有色琼斯多项式。这使得研究有色琼斯多项式的前导和拖尾系数变得有趣。在纽结上的某些条件下，似乎存在一个无限多项式，取决于纽结，其前n个系数与该纽结颜色n处的有色琼斯多项式的前n个系数一致。这些无限多项式的性质以及它们的数论性质将被研究。该项目的一部分也将找到一个更好的拓扑理解的有色琼斯多项式。为此，早期的工作的主要研究者和他的合作者将被用来解释定期琼斯多项式作为一个状态总和的子图的一个图形，嵌入在一个有向的表面，这是分配给每个结图。因此，每个状态都有三个参数，第三个参数是子图的亏格。通过在平面上的投影来研究嵌入在三维空间中的对象，例如节点，有着悠久而富有成果的传统。然而，关于原始对象的信息丢失，并且需要额外的信息来指示结的哪个弧离投影平面更远。通过在其他表面上投影，可以保留有关原始对象的更多信息。这些投影将用于理解结不变量（如琼斯多项式）的拓扑和几何性质。

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.