20/06/2017, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Louis H. Kauffman, University of Illinois at Chicago
Reconnection, Vortex Knots and the Fourth Dimension
Vortex knots tend to unravel into collections of unlinked circles by writhe preserving reconnections. We can model this unravelling by examining the world line of the knot, viewing each reconnection as a saddle point transition. The world line is then seen as an oriented cobordism of the knot to a disjoint collection of circles. Cap each circle with a disk (for the mathematics) and the world line becomes an oriented surface in four-space whose genus can be no more than one-half the number of recombinations. Now turn to knot theory between dimensions three and four and find that this genus can be no less than one-half the Murasugi signature of the knot. Thus the number of recombinations needed to unravel a vortex knot $K$ is greater than or equal to the signature of the knot $K$. This talk will review the backgrounds that make the above description intelligible and we will illustrate with video of vortex knots and discuss other bounds related to the Rasmussen invariant. This talk is joint work with William Irvine.