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 1


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.