**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.*

ruthel@math.huji.ac.il