Tonight I submitted the following paper: paper.pdf to the Journal of Knot Theory and Its Ramifications – the paper hasn’t been refereed yet so I’m merely offering it here for the reader’s enjoyment.
ON THE UNCOLOURED QUANTUM LINK INVARIANTS
ARISING FROM Uq(osp(1|2n)) AND U−q(so(2n + 1))
SACHA C. BLUMEN
School of Mathematics and Statistics,
The University of Sydney, NSW, 2006, Australia.
The Uq(osp(1|2n)) and U−q(so(2n + 1)) quantum link invariants obtained by colouring each component of a link with the (2n + 1)-dimensional irreducible representations of each quantum algebra, respectively, are the same up to a possibly obscure substitution of variables.
These quantum link invariants are identical to the Kauffman link invariants obtained from two isomorphic specialisations of the Birman-Wenzl-Murakami algebra. Each of these Kauffman link invariants is the trace of some element in the relevant specialisation of the Birman-Wenzl-Murakami algebra.
The truncated Bratteli diagrams of the semisimple quotients of these two specialisations of the Birman-Wenzl-Murakami algebra are the same, where the quotients are taken with respect to an ideal generated by a subset of elements with zero trace.
The trace of any element in one specialisation of the Birman-Wenzl-Murakami algebra is the same as the trace of the corresponding element in the other specialisation of the Birman-Wenzl-Murakami algebra up to a possibly obscure substitution of variables. This gives the corresponding result for the relevant uncoloured quantum link invariants.
Keywords: link invariants, quantum algebra, Birman-Wenzl-Murakami algebra
Mathematics Subject Classification 2000: 57M27, 17B37