20 | April | 2020 | classical knots, virtual knots, and algebraic structures related to knots

Maggie Miller (Princeton University)

Abstract: David Gabai showed that a minimum-genus surface in a knot complement remains minimum-genus when capped off into the zero-surgery on that knot, implying that surgery on a nontrivial knot can never yield $S^1\times S^2$ (thus proving the Property R conjecture). I study this problem for links. In particular, I show that if $L$ is a 2-component link with nonzero linking number and nondegenerate Thurston norm on its complement, then there exists a finite set $E\subset H_2(S^3\setminus\nu(L),\partial;\mathbb{R})$ so that if $S$ is norm-minimizing and not in $E$ up to scalar multiplication, then $\hat{S}$ is norm-minimizing in the 3-manifold obtained from $S^3$ by doing Dehn surgery on $L$ according to $\partial S$ . (The result generally holds for $n>1$ -component links with $E$ $(n-2)$ -dimensional.) The proof involves constructing a taut foliation on $S^3\setminus\nu(L)$ with nice boundary properties, motivated by Gabai’s proof of the Property R conjecture.