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Fibered Transverse Knots and the Bennequin Bound
- Publication Year :
- 2008
-
Abstract
- We prove that a nicely fibered link (by which we mean the binding of an open book) in a tight contact manifold $(M,\xi)$ with zero Giroux torsion has a transverse representative realizing the Bennequin bound if and only if the contact structure it supports (since it is also the binding of an open book) is $\xi.$ This gives a geometric reason for the non-sharpness of the Bennequin bound for fibered links. We also note that this allows the classification, up to contactomorphism, of maximal self-linking number links in these knot types. Moreover, in the standard tight contact structure on $S^3$ we classify, up to transverse isotopy, transverse knots with maximal self-linking number in the knots types given as closures of positive braids and given as fibered strongly quasi-positive knots. We derive several braid theoretic corollaries from this. In particular. we give conditions under which quasi-postitive braids are related by positive Markov stabilizations and when a minimal braid index representative of a knot is quasi-positive. In the new version we also prove that our main result can be used to show, and make rigorous the statement, that contact structures on a given manifold are in a strong sense classified by the transverse knot theory they support.<br />Comment: 18 pages, 1 figure
- Subjects :
- General Mathematics
010102 general mathematics
Skein relation
Fibered knot
Geometric Topology (math.GT)
Tricolorability
01 natural sciences
Mathematics::Geometric Topology
Knot theory
Combinatorics
Mathematics - Geometric Topology
Knot (unit)
Knot invariant
Mathematics - Symplectic Geometry
0103 physical sciences
Braid
FOS: Mathematics
Symplectic Geometry (math.SG)
010307 mathematical physics
0101 mathematics
Mathematics
Trefoil knot
Subjects
Details
- Language :
- English
- Database :
- OpenAIRE
- Accession number :
- edsair.doi.dedup.....b743651e0eccf9dd4ffbc36f8ffad969