Methods for Calculating Quantum Invariants

Abstract

The quantum SU _q (2) 3-manifold invariants associated with a primitive 4r ^th root of unity, described in the previous chapter, are fairly new and mysterious. Their use has so far been exceedingly limited in knot theory and in 3-manifold theory. Certainly they do distinguish many pairs of 3-manifolds, even pairs with the same homotopy type, but that has usually been more simply achieved by other means. However, there exist pairs of distinct manifolds with the same invariants for all r (see [85], [55] and [62]). For some manifolds, for some values of r the invariant is known by direct calculation to be zero. Superficially it might seem to be almost impossible to calculate any of these invariants. The calculation, from first principles, of the invariant corresponding to a 4r ^th root of unity involves taking an (r-2)-parallel of a surgery link giving the 3-manifold. If the link’s diagram has n crossings, that of the parallel has n(r - 2)² crossings; calculating a Jones polynomial by naive means soon becomes impractical when many crossings are involved. It will be shown here that it is in principle fairly easy to give a formula, as a summation, for the invariants of lens spaces and, more generally, for certain Seifert fibrations. Although in theory any of the invariants can always be calculated, it is sensible to use various simplifying procedures whenever possible. Some of those will be described in this chapter. Tables of specific computer calculations appear in [104] and in [62], where one can search for patterns in the resulting lists of complex numbers.