Thurston's Geometrization conjecture prompted the study of one of the main objects in topology, a manifold, from a new perspective: using geometry. Soon it was noticed that hyperbolic manifolds formed the largest and the least understood class of manifolds. Informally, a hyperbolic 3-manifold is an object modeled locally on 3-dimensional space that has hyperbolic metric. An important subclass of such manifolds are knot and link complements in 3-sphere. Additionally, knots and links are an object of study on their own, and knot theory has a number of applications in pure mathematics, as well as in applied fields of study. A knot is easy to draw as a planar diagram with information about underpasses/overpasses at crossings. However, establishing the connection between the diagram and geometric properties of the corresponding 3-manifold is a hard task. This is the first main goal of the proposal. The other goal is exploring the relation between geometric invariants and invariants that come from other areas of mathematics. The PI is engaged in mentoring student research and developing software that helps to use the geometric perspective, and will continue these activities.
With M. Thistlethwaite, the PI made progress towards the first goal by suggesting a new method for computing the hyperbolic structure directly from a link diagram. With W. Neumann, she generalized this approach to parameterize hyperbolic structure of a cusped 3-manifold. This project blends the new method with various techniques for a systematic study of the intrinsic geometry of hyperbolic links and its connection with a combinatorial picture. Among the open problems considered are questions about the canonical cell decomposition, hyperbolic volume, lengths of various arcs, geometric triangulations, etc. While these questions are interesting a priori, a significant part of the project is concerned with immediate applications of the geometric insight obtained. For example, the PI will explore the relation between geometric and quantum link invariants, investigate the connection between the geometry and arithmetic invariants of hyperbolic 3-manifolds, approach tangle tabulation using geometry and computer calculations, and tackle some other open questions.
