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 S¹. We prove that K is rationally fibered, that is, the knot complement admits a fibration over S¹. As part of the proof, we show that if K C Y has a Dehn surgery to S¹ x S², then K is rationally fibered. In the case that K admits some S¹ x S² surgery, K is Floer simple, that is, the rank of HFK(Y,K) is equal to the order of H₁(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 b₁ > 0).

Files

Additional details