Some computations of Ohtsuki series

Nori Jacoby and Ruth Lawrence

Abstract: Using the R-matrix formulation of the sl_3 invariant of links, we compute the coloured sl_3 generalised Jones polynomial for the trefoil. From this, the PSU(3) invariant of the Poincar\'e homology sphere is obtained. This takes complex number values at roots of unity. The result obtained is formally an infinite sum, independent of the order of the root of unity, which at roots of unity reduces to a finite sum. This form enables the derivation of the PSU(3) analogue of the Ohtsuki series for the Poincar\'e homology sphere, which it was shown by Thang Le could be extracted from the PSU(N) invariants of any rational homology sphere.

Keywords: Ohtsuki series, coloured Jones polynomial, quantum groups, trefoil knot, quantum invariants.

AMS subject classification: 57M27 05A30 11B65 17B37 57R56

Length: 18 pages

Reference: NATO Advanced Research Workshop, in `Advances in Topological Quantum Field Theory', Kluwer (2004) 53-70 MR2147416 (2006b:57015)

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Last updated April 15th, 2018.