Reconnection Numbers of Knotted Vortices

Abstract: Knotted vortices such as those produced in water by Kleckner and Irvine tend to transform by reconnection to collections of unknotted and unlinked circles. The reconnection number $R(K)$ of an oriented knot of link $K$ is the least number of reconnections (oriented re-smoothings) needed to unknot/unlink $K$. Putting this problem into the context of knot cobordism, we show, using Rasmussen's Invariant that the reconnection number of a positive knot is equal to twice the genus of its Seifert spanning surface. In particular an (a,b) torus knot has $R=(a−1)(b−1)$. For an arbitrary unsplittable positive knot or link $K$, $R(K)=c(K)−s(K)+1$ where $c(K)$ is the number of crossings of K and $s(K)$ is the number of Seifert circles of $K$. Examples of vortex dynamics are illustrated in the talk.

geometric topology

Audience: researchers in the topic

( video )

GEOTOP-A seminar

Series comments: Web-seminar series on Applications of Geometry and Topology