Orthogonal Quantum Group Invariants of Links

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.