# Dehn surgery on links vs the Thurston norm

## 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.

References:

Miller, M., The effect of link Dehn surgery on the Thurston norm. https://arxiv.org/abs/1906.08458

## 3 thoughts on “Dehn surgery on links vs the Thurston norm”

1. Micah (bounces to Maggie Miller) says:

1. chrisman.76 says:

Very cool! Is there is a way to characterize the exceptional set $E$ for different link properties (e.g. hyperbolic, alternating, fibered…)?

1. Maggie Miller says:

Not directly — the exceptional set is determined by the shape of the unit-ball of the Thurston norm on the link complement. So for example, if a link has two components, then the unit ball is bounded by a polygon in the plane. Each face of the polygon has an area, which is the area of the smallest lattice parallelogram with edges through the corners of the face. The set E contains slopes for:

Each corner of the unit ball,
Each midpoint of a face of the unit ball,
Extra slopes depending on the area of each face.

So basically, if the unit ball is simpler (has fewer faces), then we expect the exceptional set to be smaller.

It’s easier to compute the unit ball for alternating links, so you might be able to say something about that case. But in general, I don’t know how to directly relate the geometry of the link complement to the size of $E$. (We’re sort of seeing two different versions of exceptional surgeries here — we’re studying exceptional behavior of the Thurston norm rather than exceptional surgeries in terms of changing geometry). It would be interesting to know if you could say something more specific, though!