# The effect of link Dehn surgery on the Thurston norm

## TL;DR

This paper investigates how Dehn surgery on links in rational homology spheres affects the Thurston norm of embedded surfaces, providing conditions under which the norm-minimizing property is preserved.

## Contribution

It characterizes when Thurston norm-minimizing surfaces remain minimal after Dehn filling, especially for two-component links in rational homology spheres.

## Key findings

- Capped-off surfaces remain norm-minimizing outside a finite set of homology rays.
- Provides bounds on surgeries yielding $S^1\times S^2$ in integer homology spheres.
- Uses techniques from Gabai's proof of the Property R conjecture.

## Abstract

Let $L$ be an $n$-component link ($n>1$) with pairwise nonzero linking numbers in a rational homology $3$-sphere $Y$. Assume the link complement $X:=Y\setminus\nu(L)$ has nondegenerate Thurston norm. In this paper, we study when a Thurston norm-minimizing surface $S$ properly embedded in $X$ remains norm-minimizing after Dehn filling all boundary components of $X$ according to $\partial S$ and capping off $\partial S$ by disks. In particular, for $n=2$ the capped-off surface is norm-minimizing when $[S]$ lies outside of a finite set of rays in $H_2(X,\partial X;\mathbb{R})$. When $Y$ is an integer homology sphere this gives an upper bound on the number of surgeries on $L$ which may yield $S^1\times S^2$. The main techniques come from Gabai's proof of the Property R conjecture and related work.