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

Let $L$ be a link and ${\mathrm{\Phi }}_L^A(q)$ 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 ${\mathrm{\Phi }}_L^{osp(1\mathrm{\mid }2n)}(q)=F_L(-q^{2n},q)$ and $\Phi^{so(2n+1)}{L}(-q) = F{L}(q^{2n},-q)$foreachpositiveinteger$n$andallsufficientlylarge$f$, and that ${\mathrm{\Phi }}_L^{osp(1\mathrm{\mid }2n)}(q)$ and ${\mathrm{\Phi }}_L^{so(2n+1)}(-q)$ are identical up to a substitution of variables. For at least one class of links $F_L(-r,-s)=F_L(r,s)$ implying $\Phi^{osp(1|2n)}{L}(q) = \Phi^{so(2n+1)}{L}(-q)$ for these links.