## On the $U_{q}(osp(1|2n))$ and $U_{-q}(so(2n+1))$ uncoloured quantum link invariants

January 2, 2009

I’ve resubmitted my paper on the $U_{q}(osp(1|2n))$ AND $U_{-q}(so(2n+1))$ uncoloured link invariants to the Journal of Knot Theory and its Ramifications. Its abstract is below (in latex).

Update: (3/1/09) the paper has been accepted for publication.

Update: (7/1/09) a preprint of the paper has been published on the website of the School of Mathematics and Statistics, University of Sydney.

On the $U_{q}(osp(1|2n))$ and $U_{-q}(so(2n+1))$ uncoloured quantum link invariants

\begin{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.
\end{abstract}

## Job to do today

August 10, 2007

I’m having a day for my own work rather than for my employer’s work, and I’d like today to finish editing the paper I submitted to the Journal of Knot Theory – I have the referee’s very helpful comments and I now have the time to consider the comments and make the appropriate changes to the paper.

The academic world is funny. To have a chance of getting into it you really have to have quite a few published papers – but most people, unless they’re already in the academic world, don’t have the time to write papers!

Time for coffee. Then I’ll start on the editing.