Null surgery on knots in L-spaces

Abstract

Let $K$ be a knot in an L-space $Y$ with a Dehn surgery to a surface bundle over $S^1$. We prove that $K$ is rationally fibered, that is, the knot complement admits a fibration over $S^1$. As part of the proof, we show that if $K\subset Y$ has a Dehn surgery to $S^1 \times S^2$, then $K$ is rationally fibered. In the case that $K$ admits some $S^1 \times S^2$ surgery, $K$ is Floer simple, that is, the rank of $\hat{HFK}(Y,K)$ is equal to the order of $H_1(Y)$. By combining the latter two facts, we deduce that the induced contact structure on the ambient manifold $Y$ is tight. In a different direction, we show that if $K$ is a knot in an L-space $Y$, then any Thurston norm minimizing rational Seifert surface for $K$ extends to a Thurston norm minimizing surface in the manifold obtained by the null surgery on $K$ (i.e., the unique surgery on $K$ with $b_1>0$).