Invariants for knots, and graphs on surfaces

体积猜想声称在一侧的纽结的索的琼斯多项式和另一侧的纽结补的双曲体积之间存在深刻的关系。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，以及它与结补的几何关系。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。在某些条件下，纽结上有一个幂级数，它决定了它的有色琼斯多项式的前k个系数，对于每一个固定的k和足够大的颜色。研究人员将研究这些幂级数的几何和数论性质。此外，在早期的工作中的调查员和他的合作者的琼斯多项式被解释为一个状态总和的子图的图，嵌入在一个有向的表面上，这是分配给每个结图。当研究嵌入在三维空间中的物体时，例如节点，通过它们在平面上的投影，关于原始物体的信息丢失，并且需要额外的信息来指示节点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。

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.