# Transverse knots with knot types of strongly quasi-positive knots

In 2008, Etnyre and Van Horn-Morris proved that if $$L$$ is a fibered strongly quasi-positive link, there is a unique (up to transverse isotopy) transverse link with the knot type of $$L$$ in the standard tight contact structure of $$S^3$$ with maximal self-linking number.

What is known for (not necessarily fibered) strongly quasi-positive links? If $$L$$ is strongly quasi-positive, is there a classification of transverse links with the knot type of $$L$$, with maximal self-linking number? Any comments or references would be appreciated.

## 1 Answer

The twist knot $$K_{-6} = m(7_2)$$ is strongly quasipositive, not fibered, genus-1, and it's Legendrian non-simple; it has five Legendrian isotopy classes with Thurston-Bennequin number 1, rotation number 0. A good source of material for low-crossing knots is the Legendrian knot atlas, by Chongchitmate and Ng.

Legendrian twist knots have been classified by Etnyre, Ng, and Vértesi (J. Eur. Math. Soc. 15 (2013)). I doubt that there exists a classification for arbitrary strongly quasipositive knots.