Examples of different 3-manifolds with the same invariants of Witten and Reshetikhin–Turaev

Abstract

THE AIM of this paper is to provide examples of different 3-manifolds that have the same Reshetikhin-Turaev invariants z,, for all integers r. These invariants were studied by Kirby and Melvin [9]. They slightly modified them so that the resulting invariants r,(M) conjugate under orientation reversal and appear to be exactly the same as Witten’s invariants of M (see [19]) with the canonical 2-framing of Atiyah. Moreover, they found a formula for t,(M) in terms of the classical Jones polynomials of cables of L. The simplest examples are obtained by n-surgery on a pair of mutant knots (Corollary 3.4), where n must be chosen large enough (it is not always obvious for a given knot how large n must be). With a little more work we obtain examples of pairs of knots for which all p/q-Dehn surgeries produce distinct pairs of 3-manifolds with the same r, invariants for all I (Corollary 5.6). These pairs are K # K and K # -K for a nonreversible, hyperbolic knot K, e.g. 8,,. Since K # K and K # -K, and their (p, q) cables have the same knot polynomials (i.e., Jones, Homfly or Kauffman) [14], then their corresponding 3-manifolds should also not be distinguishable by more general 3-manifold invariants based on current knot polynomials.