One central question left unanswered by Perelman's geometrization theorem is exactly how the combinatorial features of a 3-manifold should relate to its geometry. The principal investigators Futer and Purcell will study several aspects of this question. The first goal of this project is to use the combinatorial complexes associated a surface to give explicit estimates on the geometry of a fibered hyperbolic 3-manifold. A second, closely related, goal is to use braid presentations of a generic knot or link in order to give explicit, diagrammatic estimates on the volume of its complement. Third, the PIs will continue their joint project with Kalfagianni to relate the geometric topology of knot and link complements to quantum invariants such as the colored Jones polynomials. Finally, the investigators will continue their joint work with Cooper to understand the geometric properties of unknotting tunnels.
A 3-manifold is a space where an object such as a helicopter can move around in three distinct perpendicular directions. The universe that we inhabit is a 3-manifold whose global geometry we do not yet understand. Another rich source of examples comes from the spaces that surround different knots. Powerful theorems of Thurston, Perelman, and Mostow imply that almost every knot complement, and more generally almost every 3-manifold, has a unique hyperbolic metric. That is, there is a standard way to measure the space, so that every 2-dimensional cross-section curves like a saddle. At the moment, while we know that this standard hyperbolic metric exists, very little is known about how to relate it to easily computable quantities such as the complexity of a knot diagram. The main goal of this project is to make these relations much more concrete. One important feature of this project is that the geometric problems studied by the PIs are very visual and hands-on, with natural spin-offs into both software applications and projects for students.
Take a string, knot it up, and then fuse the ends together. Such objects are the study of mathematical knot theorists, who wish to give mathematical conditions to answer questions such as when the knot can be "untied" without cutting the string, or what happens when a single crossing is changed, or how to simplify the knot to a recognizable form. Often it is easier to approach these questions not by studying the knot, but by investigating all of space except the knot. The resulting space, called the knot complement, has important geometric properties. It often admits a hyperbolic metric, meaning that the space is negatively curved with constant negative curvature. Such spaces, as well as closely related spaces, were the objects of study of this project. In general, it is a difficult problem to relate properties of a diagram describing the knot to the hyperbolic geometry of the corresponding knot complement. We investigated this problem from several directions in this project. Among other results, we determined conditions on the knot diagram that guarantee the resulting knot complement has bounded hyperbolic volume, with bounds coming from the diagram. We also determined information on certain shortest paths, or geodesics, through these spaces. In addition to the intellectual merit of determining geometric properties of knots and related spaces, this project had a broader impact, in that it provided training and research experiences for undegraduate and graduate students. The results have appeared in various peer-reviewed mathematical journals, and have also been distributed through preprints and talks concerning the work.
