The PSU(3) invariant of the Poincaré homology sphere

Using the R-matrix formulation of the sl3 invariant of links, we compute the coloured sl3 generalised Jones polynomial for the trefoil. From this, the PSU(3) invariant of the Poincaré 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 Poincaré homology sphere, which it was shown by Thang Le could be extracted from the PSU(N) invariants of any rational homology sphere.