Volume 6, issue 1 (2006)

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Legendrian knots and monopoles

Tomasz S Mrowka and Yann Rollin

Algebraic & Geometric Topology 6 (2006) 1–69
 arXiv: math.DG/0410559
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 $M$.

Keywords
contact structures, Legendrian knots, Bennequin inequality, excision, monopoles
Mathematical Subject Classification 2000
Primary: 57R17, 57M25, 57M27, 57R57
Publication
Accepted: 10 July 2005
Published: 24 February 2006
Authors
 Tomasz S Mrowka MIT 77 Massachusetts Avenue Cambridge MA 02139 USA Yann Rollin Imperial College Huxley Building 180 Queen’s Gate London SW7 2AZ UK