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

TL;DR

This paper establishes explicit relationships between certain quantum superalgebra link invariants and Kauffman link invariants, revealing their equivalence under specific parameter substitutions for large algebra ranks.

Contribution

It demonstrates that the quantum superalgebra invariants for $osp(1|2n)$ and $so(2n+1)$ are directly related to Kauffman invariants, providing explicit formulas and conditions for their equivalence.

Findings

$ ext{Phi}^{osp(1|2n)}_{L}(q) = F_{L}(-q^{2n},q)$ for all links and large $f$

$ ext{Phi}^{so(2n+1)}_{L}(-q) = F_{L}(q^{2n},-q)$ for all links and large $f$

For some links, $ ext{Phi}^{osp(1|2n)}_{L}(q) = ext{Phi}^{so(2n+1)}_{L}(-q)$

Abstract

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.