Legendrian knots and monopoles

Abstract

We prove a generalization of Bennequin’s inequality for Legendrian knots in a 3-dimensional contact manifold $\left(Y,\xi \right)$, under the assumption that $Y$ is the boundary of a 4-dimensional manifold $M$ and the version of Seiberg-Witten invariants introduced by Kronheimer and Mrowka [Invent. Math. 130 (1997) 209–255] is nonvanishing. The proof requires an excision result for Seiberg-Witten moduli spaces; then the Bennequin inequality becomes a special case of the adjunction inequality for surfaces lying inside $M$.