6-28【Tao Su】Log-concavity of invariants from Legendrian knot theory and character varieties

Basing on examples, we propose a conjecture on the log-concavity of ruling polynomials (Legendrian analogue of Jones polynomials) for Legendrian links. In the case of Legendrian torus knots, we prove the conjecture via a connection to the HOMFLY-PT polynomials. Moreover, we explain some algebraic geometry behind via some 'BPS calculus', in the case of Legendrian links arising from plane algebraic curve singularities. Finally, there is a natural generalization of the conjecture to character varieties. Based on work in progress.