Invariants for knots, and graphs on surfaces

近年来，Jones多项式等多项式结不变量的研究有了新的发展。特别是体积猜想，声称一边结的电缆的琼斯多项式和另一边结的双曲体积之间的深刻关系，导致了一个新的观点。该项目的主题受到了体积猜想的启发。范围是为了更好地理解有色琼斯多项式和双曲体积。在首席研究员和他的合作者的早期工作中，证明了某些类型的结的双曲体积的界限可以从彩色琼斯多项式的系数中读出。这使得研究有色琼斯多项式的前导和后导系数变得很有趣。在结的特定条件下，似乎有一个无限多项式，取决于结，它的前n个系数与结的颜色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。项目的一部分还将是找到对有色琼斯多项式更好的拓扑理解。为此，将使用主要研究者和他的合作者的早期工作，将常规琼斯多项式解释为嵌入在定向表面上的图的子图的状态和，该子图被分配给每个结图。因此，每个状态都配备了三个参数，第三个参数是子图的属。它有着悠久而富有成果的传统，通过在平面上的投影来研究嵌入三维空间的物体，例如结。然而，关于原始对象的信息丢失了，并且需要额外的信息来指示结的哪条弧离投影平面更远。通过在其他表面上投影，可以保留原始物体的更多信息。这些投影将用于理解像琼斯多项式这样的结不变量的拓扑和几何性质。

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.