Null surgery on knots in L-spaces

Abstract

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