Advances in Theoretical and Mathematical Physics

Volume 24 (2020)

Number 8

The second of two special issues in honor of Cumrun Vafa’s 60th birthday

Higher genus knot contact homology and recursion for colored HOMFLY-PT polynomials

Pages: 2067 – 2145



Tobias Ekholm (Department of Mathematics, Uppsala University, Uppsala, Sweden; and Institut Mittag-Leffler, Djursholm, Sweden)

Lenhard Ng (Department of Mathematics, Duke University, Durham, North Carolina, U.S.A.)


We sketch a construction of Legendrian Symplectic Field Theory (SFT) for conormal tori of knots and links. Using large $N$ duality and Witten’s connection between open Gromov–Witten invariants and Chern–Simons gauge theory, we relate the SFT of a link conormal to the colored HOMFLY‑PT polynomials of the link. We present an argument that the HOMFLY‑PT wave function is determined from SFT by induction on Euler characteristic, and also show how to, more directly, extract its recursion relation by elimination theory applied to finitely many noncommutative equations. The latter can be viewed as the higher genus counterpart of the relation between the augmentation variety and Gromov–Witten disk potentials established in [1] by Aganagic, Vafa, and the authors, and, from this perspective, our results can be seen as an SFT approach to quantizing the augmentation variety.

T.E. is supported by the Knut and Alice Wallenberg Foundation and by the Swedish Research Council. Parts of the paper are based upon work supported by the National Science Foundation under Grant No. DMS-1440140 while the author was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the 2018 spring semester.

L.N. is partially supported by NSF grants DMS-1406371 and DMS-1707652.

Published 28 September 2021