On the U_q(osp(1|2n)) and U_−q(so(2n + 1)) uncoloured quantum link invariants

Sacha C. Blumen

Abstract

Let L be a link and Φ_L^A(q) its link invariant associated with the vector representation of the quantum (super)algebra U_q(A). Let F_L(r,s) be the Kauffman link invariant for L associated with the Birman-Wenzl-Murakami algebra BWM_f(r,s) for complex parameters r and s and a sufficiently large rank f. For an arbitrary link L, we show that Φ_L^osp(1|2n)(q) = F_L(−q²ⁿ,q) and Φ_L^so(2n+1)(−q) = F_L(q²ⁿ,−q) for each positive integer n and all sufficiently large f, and that Φ_L^osp(1|2n)(q) and Φ_L^so(2n+1)(−q) are identical up to a substitution of variables. For at least one class of links F_L(−r,−s) = F_L(r,s)) implying Φ_L^osp(1|2n)(q) = Φ_L^so(2n+1)(−q) for these links.

Keywords: Quantum superalgebra, link, Kauffman invariant.

AMS Subject Classification: Primary 57M27; secondary 17B37.