Geometry & Topology

Knot distortion is a geometric knot invariant introduced by Gromov in the 1980s. It is the smallest bi-Lipschitz constant required to embed the knot in $\mathbb R^3$. The most basic question about distortion is: how do we estimate it? Although distortion has a simple definition, this question has been surprisingly difficult.

In quantitative topology, we ask refinements of topological questions that include the sizes of the objects involved. Estimating knot distortion appears to be such a question, as distortion is a way to measure the complexity of a knot. However, it is not clear what should be the qualitative topological statements that underlie distortion statements.

In this talk, I'll explain how to build a certain type of combinatorial world, in which these underlying qualitative statements live. The objects of this world are constructed from (isotopy classes of) embedded curves on surfaces. I'll explain how statements about knot distortion can be translated into statements in the combinatorial world, and how to leverage this to get lower bounds on distortion.