Invariants for knots, and graphs on surfaces

近年来，像琼斯多项式这样的多项式纽结不变量的研究获得了新的势头。特别地，声称一边纽结的缆索的琼斯多项式与另一边纽结补的双曲体积之间存在深刻关系的Volume猜想引出了一个新的观点。该项目的主题是受体积猜想的启发。这个范围是为了更好地理解有色琼斯多项式和双曲体积。在主要研究人员和他的合作者的早期工作中，证明了某些类型的纽结的双曲体积的界可以从有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系数和拖尾系数变得很有趣。在某些条件下，纽结上似乎有一个无穷多项式，这取决于纽结，它的前n个系数与该纽结颜色为n的有色琼斯多项式的前n个系数一致。我们将研究这些无穷多项式的性质以及它们的数论性质。该项目的一部分还将是找到对有色琼斯多项式的更好的拓扑理解。为此，将使用主要研究人员及其合作者的早期工作，该工作将正则琼斯多项式解释为嵌入在定向表面上的图的子图上的状态和，该图被分配给每个结图。因此，每个状态都有三个参数，第三个参数是子图的亏格。通过它们在平面上的投影来研究嵌入在三维空间中的对象，例如结，有着悠久而卓有成效的传统。但是，有关原始对象的信息会丢失，并且需要其他信息来指示结的哪个圆弧距离投影平面更远。通过在其他表面上投影，可以保留关于原始对象的更多信息。这些投影将用于了解纽结不变量的拓扑和几何性质，如琼斯多项式。

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.