On the $U_q(osp(1|2n))$ and $U_{-q}(so(2n+1))$ Uncoloured Quantum Link Invariants

Abstract

Let $L$ be a link and ${\Phi }_L^A(q)$ 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 $BW{M}_f(r,s)$ for complex parameters $r$ and $s$ and a sufficiently large rank $f$. For an arbitrary link $L$, we show that ${\Phi }_L^{osp(1|2n)}(q)=F_L(-q^{2n},q)$ and ${\Phi }_L^{so(2n+1)}(-q)=F_L(q^{2n},-q)$ for each positive integer $n$ and all sufficiently large $f$, and that ${\Phi }_L^{osp(1|2n)}(q)$ and ${\Phi }_L^{so(2n+1)}(-q)$ are identical up to a substitution of variables. For at least one class of links $F_L(-r,-s)=F_L(r,s)$ implying ${\Phi }_L^{osp(1|2n)}(q)={\Phi }_L^{so(2n+1)}(-q)$ for these links.

16 pages, 4 figures, accepted for publication by the Journal of Knot Theory and its Ramifications