Orthogonal quantum group invariants of links

Lin Chen and Qingtao Chen

Vol. 257 (2012), No. 2, 267–318

DOI: 10.2140/pjm.2012.257.267

Abstract

We first study the Chern–Simons partition function of orthogonal quantum group invariants and then propose a new orthogonal Labastida–Mariño–Ooguri–Vafa (LMOV) conjecture as well as a degree conjecture for free energy associated to the orthogonal Chern–Simons partition function. We prove the degree conjecture and some interesting cases of the orthogonal LMOV conjecture. In particular, we provide a formula of the colored Kauffman polynomials for torus knots and links, and applied this formula to verify certain cases of the conjecture at roots of unity except 1. We also derive formulas of Lickorish–Millett type for Kauffman polynomials and relate all these to the orthogonal LMOV conjecture.

Keywords

quantum invariant

Mathematical Subject Classification 2010

Primary: 57M27, 81R50

Milestones

Received: 22 June 2011

Revised: 8 November 2011

Accepted: 14 November 2011

Published: 4 July 2012