Topology of Vortex Reconnection

TL;DR

This paper links vortex reconnection in knotted vortices to knot theory, showing the reconnection number equals twice the genus for positive knots and providing formulas for specific cases, with implications for vortex dynamics.

Contribution

It establishes a mathematical relationship between vortex reconnection numbers and knot invariants, using Rasmussen's Invariant to connect topology with fluid dynamics.

Findings

Reconnection number of positive knots equals twice the genus of their Seifert surface.

For torus knots, the reconnection number is (a-1)(b-1).

Reconnection number for positive knots is c(K) - s(K) + 1.

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 paper.