Geometric Invariants of Knots

Given two arbitrary knots (tangled up strings with their ends tied together), how can we (easily) tell if they are different or not? In general, this problem is extremely difficult to answer, and has led to the development of a variety of knot invariants. In this talk, we will examine geometric invariants of knots. These are knot invariants that arise from examining the (often hyperbolic) geometry of the space surrounding a knot. Historically, these invariants have been extremely useful in helping classify knots. However, it is possible to construct large sets of knots that are geometrically similar, that is, all the knots in such a set are different, yet these knots have a number of geometric invariants in common. We will give two examples of geometrically similar sets of knots and raise some interesting questions about geometric invariants.