On knots in overtwisted contact structures

Details On knots in overtwisted contact structures On knots in overtwisted contact structures

2010-12-16

arxiv.org/abs/1012.3745v2

Mathematics

Symplectic Geometry

23 pages, 3 figures, exposition clarified at several points and more background provided

Scientific paper

We prove that each overtwisted contact structure has knot types that are represented by infinitely many distinct transverse knots all with the same self-linking number. In some cases, we can even classify all such knots. We also show similar results for Legendrian knots and prove a "folk" result concerning loose transverse and Legendrian knots (that is knots with overtwisted complements) which says that such knots are determined by their classical invariants (up to contactomorphism). Finally we discuss how these results partially fill in our understanding of the "geography" and "botany"' problems for Legendrian knots in overtwisted contact structures, as well as many open questions regarding these problems.

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