Recursion formulas for HOMFLY and Kauffman invariants (Q2878657)

Embeddings of simple Lie algebras which are consistent with embeddings of their Dynkin diagrams induce embeddings of the corresponding quantized universal enveloping algebras. In the case of classical Lie algebras, we have the following:NEWLINENEWLINE\((0.1): U_q (sl_{n-k})\otimes U_q(sl_k)\subset U_q(sl_n),\)NEWLINENEWLINE\((0.2): U_q (sl_{k})\otimes U_q(so_{2n-2k})\subset U_q(so_{2n}), U_q (sl_{k})\otimes U_q(sp_{2n-2k})\subset U_q(sp_{2n})\)NEWLINENEWLINE\((0.3): U_q (sl_{k})\otimes U_q(so_{2n-2k+1})\subset U_q(so_{2n+1}),\)NEWLINENEWLINE\((0.4): U_q (sl_{n})\subset U_q(so_{2n}), U_q (sl_{n})\subset U_q(sp_{2n}), U_q (sl_{n})\subset U_q(so_{2n+1}). \)NEWLINENEWLINEIn Section 1, starting from the tangle category \(\underline{Tan}_{{\mathbb C}[t^{\pm 1},q^{\pm 1}]}\), the authors construct a covariant functor \(\phi_{q;t,q}\) from the braided monoidal category \({\mathcal H}_{q;tq}\) (HOMFLY skein modules) to the braided monoidal category \({\mathcal H}_{q;t,q}\). The functor \(\phi_{q;t,q}\) gives a recursive relation for HOMFLY polynomials of framed links. The recursion corresponds to the embedding \((0.1)\) with \(k=1\).NEWLINENEWLINEThis recursion formula is extended to all \(k\) for HOMFLY polynomials by constructing the corresponding covariant braided monoidal functor from one skein category to another (Section 2), see also \textit{F. Jaeger} [Enseign. Math. (2) 35, No. 3--4, 323--361 (1989; Zbl 0705.57004)]NEWLINENEWLINEIn Section 3, the authors describe the recursion relations corresponding to the embeddings \((0.4)\) in a similar way.NEWLINENEWLINEIn Section 4, the authors define the skein category \({\mathcal K}_{q,s}\) corresponding to one of the classical Lie algebras \(so_{2n+1},so_{2n}\) or \(sp_{2n}\) and the skein category \({\mathcal HK}_{q;s,t}\) corresponding to products of HOMFLY and Kauffman invariants and describe the functor \( \chi _{q;s,t}: {\mathcal K}_{q,st^2}\to {\mathcal HK}_{q;s,t}\). This gives the recursion formula for Kauffman polynomial corresponding to embeddings \((0.2)\) and \((0.3)\) which is the main result of the given paper.