**The PSU(3) invariant of the Poincare homology sphere**

**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 Poincare 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 Poincare homology
sphere, which it was shown by Thang Le could be extracted from the
*PSU(N)* invariants of any rational homology sphere.

**Keywords: **Manifold invariants, quantum groups, TQFT, perturbative
expansion.

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

**Length: **16 pages

**Reference: ** *Topology and Its Applications***127 **(2003)
153-168

MR1953324 (2003m:57032) (review by * Razvan Gelca *.)

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*Last updated February 21st, 2001.*

ruthel@math.huji.ac.il