The effect of link Dehn surgery on the Thurston norm

Abstract

Abstract Let $L$ be an $n$-component link ($n>1$) with pairwise nonzero linking numbers in a rational homology $3$-sphere $Y$. Assume the complement $X:=Y\setminus \nu (L)$ has nondegenerate Thurston norm. 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 $H_2(X,\partial X;{\mathbb {R}})$, answering a conjecture of Gabai. 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.