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Legendrian ribbons in overtwisted contact structures. (English) Zbl 1229.57003

Summary: We show that a null-homologous transverse knot \(K\) in the complement of an overtwisted disk in a contact 3-manifold is the boundary of a Legendrian ribbon if and only if it possesses a Seifert surface \(S\) such that the self-linking number of \(K\) with respect to \(S\) satisfies \(\text{sl}(K,S)=-\chi(S)\). In particular, every null-homologous topological knot type in an overtwisted contact manifold can be represented by the boundary of a Legendrian ribbon. Finally, we show that a contact structure is tight if and only if every Legendrian ribbon minimizes genus in its relative homology class.

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
57R17 Symplectic and contact topology in high or arbitrary dimension
53D10 Contact manifolds (general theory)
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References:

[1] Baader S., Osaka J. Math. 42 pp 257–
[2] Bennequin D., Astérisque pp 83–
[3] DOI: 10.5802/aif.1288 · Zbl 0756.53017 · doi:10.5802/aif.1288
[4] DOI: 10.1016/B978-044451452-3/50004-6 · doi:10.1016/B978-044451452-3/50004-6
[5] DOI: 10.1016/S0001-8708(02)00027-0 · Zbl 1047.57006 · doi:10.1016/S0001-8708(02)00027-0
[6] DOI: 10.1016/S0040-9383(96)00035-3 · Zbl 0904.57006 · doi:10.1016/S0040-9383(96)00035-3
[7] DOI: 10.1007/s002220050245 · Zbl 0902.57007 · doi:10.1007/s002220050245
[8] DOI: 10.1016/0040-9383(92)90017-C · Zbl 0763.57008 · doi:10.1016/0040-9383(92)90017-C
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