Invariants for knots, and graphs on surfaces

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有很深的关系。这个项目的灵感来自于体积猜想。范围是为了更好地理解有色琼斯多项式，以及它与结补几何的关系。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。在某些条件下，对于每个固定的k和足够大的颜色，有一个分配给结的幂级数决定了它的有色琼斯多项式的前k个系数。研究者将研究这些幂级数的几何和数论性质。此外，在研究者和他的合作者的早期工作中，琼斯多项式被解释为一个图的子图的状态和，嵌入在一个定向表面上，分配给每个结图。研究了这些图及其子图的属与有色琼斯多项式性质的关系。当研究嵌入在三维空间中的对象时，例如结，通过它们在平面上的投影来丢失原始对象的信息，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。

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.