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

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

2009-01-21

arxiv.org/abs/0901.3232v1

Mathematics

Quantum Algebra

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

Scientific paper

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.

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