Turaev-viro Modules of Satellite Knots

Given an oriented knot K in S and a TQFT, Turaev and Viro defined modules somewhat analogous to the Alexander module. We work with the (Vp, Zp) theories of Blanchet, Habegger, Masbaum and Vogel [BHMV] for p ≥ 3, and consider the associated modules. In [G], we defined modules which also depend on the extra data of a color c which is assigned to a meridian of the knot in the construction of the module. These modules can be used to calculate the quantum invariants of cyclic branched covers of knots and have other uses. Suppose now that S is a satellite knot with companion C, and pattern P. We give formulas for the Turaev-Viro modules for S in terms of the Turaev-Viro modules of C and similar data coming from the pattern P. We compute these invariants explicitly in several examples. This version:(10 /11 /96); First Version: (10 /2 /96) §1 Turaev-Viro modules Let (V, Z) be a Topological Quantum Field Theory over a field f defined on a cobordism category whose morphisms are oriented 3-manifolds perhaps with extra structure. Let (M,χ) be a closed oriented 3-manifold M with this extra structure together with χ ∈ H(M) where χ : H1(M) → Z is onto. Let M∞ denote the infinite cyclic cover of M given by χ. Consider a fundamental domain E for the action of the integers on M∞ bounded by lifts of a surface Σ dual to χ, and in general position. E can be viewed as a cobordism from Σ to itself. Z(E) can be viewed as an endomorphism of V (Σ). Let K(V ) be the generalized 0-eigenspace for the action of Z(E) on V (Σ), i.e. K(V ) = ∪k≥1 Kernel(Z ). Z(E) induces an automorphism Z(E) of V (Σ) = V (Σ)/K(V ). Alternatively V can be defined as ∩k≥1 Image(Z ). The Turaev-Viro module (M,χ) associated to (V, Z) is simply is V (Σ) viewed as a f [t, t]-module where t acts by Z(E). Theorem 1.1 (Turaev-Viro). This module does not depend on the choice of E. Sketch of Proof. A detailed exposition of Turaev-Viro’s proof [TV] is given in [G,§1]. Here we give the main idea. Suppose E is another choice of fundamental domain. Without loss of generality we may assume that E has been shifted by the covering 1991 Mathematics Subject Classification. 57M99.