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 (Y,ξ), 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
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Publication
Received: 8 November 2005
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