Title: Estimating the virtual slice genus of classical knots in $S^3$.
Speaker: Micah Chrisman (Ohio State University)
Abstract: For a compact connected oriented surface $\Sigma$, a knot $K$ in $\Sigma \times [0,1]$ is said to be virtually slice if there is a compact connected oriented $3$-manifold $W$ and a disc $D$ smoothly embeddded in $W \times [0,1]$ such that $\partial D=K$. The virtual slice genus of $K$ is the smallest genus orientable surface that $K$ bounds in some $W \times [0,1]$, where the minimum is taken over all compact connected oriented $3$-manifolds $W$. A classical knot in $S^3$ can be considered as a knot in the thickened surface $S^2 \times [0,1]$. A conjecture of Kauffman states that the slice genus of a knot in $S^3$ is equal to its virtual slice genus. Boden and Nagel proved the conjecture to be true for slice knots in $S^3$. In this talk, we provide some positive evidence for Kauffman's conjecture in both the smooth and topological categories. In particular, we will prove Kauffman's conjecture is true for all classical knots up to crossing number 11 and for 2150 of the 2175 clasical knots with crossing number 12. The first portion of this talk will be an introduction to virtual knot concordance. This project is joint work with H. U. Boden and R. Gaudreau.
Seminar URL: https://www.coreyjonesmath.com/qaqt-seminar-osu