## 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}(−*q*^{2n},*q*) and *Φ*_{L}^{so(2n+1)}(*−q*) = *F*_{L}(*q*^{2n},−*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.