On Stable Khovanov Homology of Torus Knots

Abstract

We conjecture that the stable Khovanov homology of torus knots can be described as the Koszul homology of an explicit irregular sequence of quadratic polynomials. The corresponding Poincaré series turns out to be related to the Rogers–Ramanujan identity.

2000 AMS Subject Classification:

• 57M25
• 13D02
• 17B69
• 11P84

Keywords:

• Khovanov homology
• torus knots
• Koszul complex
• Rogers-Ramanujan identity

ACKNOWLEDGMENTS

We are grateful to B. Feigin, S. Gukov, M. Hagencamp, M. Khovanov, A. Kirillov Jr., S. Loktev, L. Rozansky, M. Stošić, J. Sussan, O. Viro, and V. Shende for useful discussions. Special thanks to A. Shumakovitch for providing us with valuable Khovanov homology data and explaining the Conjecture 1.8. Most of the computations of the Koszul homology were done using Singular, a computer algebra system. The research of E. G. was partially supported by grants RFBR-10-01-00678, NSh-8462.2010.1 and the Simons foundation. The research of A. O. was partially supported by the NSF and the Sloan Foundation.

To Sofia, Tommy, and Anna.

Notes

¹Available at http://www.singular.uni-kl.de .