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

Let $L$ be a link and $\Phi^{A}_{L}(q)$ 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 $\Phi^{osp(1|2n)}_{L}(q) = F_{L}(-q^{2n},q)$ and $\Phi^{so(2n+1)}_{L}(-q) = F_{L}(q^{2n},-q)$ for each positive integer $n$ and all sufficiently large $f$, and that $\Phi^{osp(1|2n)}_{L}(q)$ and $\Phi^{so(2n+1)}_{L}(-q)$ are identical up to a substitution of variables. For at least one class of links $F_{L}(-r,-s) = F_{L}(r,s)$ implying $\Phi^{osp(1|2n)}_{L}(q) = \Phi^{so(2n+1)}_{L}(-q)$ for these links.