Fox's congruence classes and the quantum-\(SU(2)\) invariants of links in 3-manifolds
For non-negative integers \(n,q\), two tame oriented framed links in a closed connected oriented \(3\)-manifold \(M\) are called \((n,q)\)-congruent if they differ by a sequence of full \(n\)-twists along disks whose oriented intersection numbers with the corresponding link are divisible by \(q\). This notion does generalize \textit{R. H. Fox}'s definition of conguence of knots in \(S^3\) [Osaka Math. J. 10, 37-41 (1958; Zbl 0084.19204)]. The author proves the following results: Let \(n>1\) be integer and let \(A\) be a primitive \(4n\)-th root of unity. If framed links \(K,L\) in \(M\) are \((n,2)\)-congruent or \((4n,1)\)-congruent then their \(SU(2)\)-quantum invariants at \(A\) are equal. (The definition of the quantum invariant is given e.g. in [\textit{W. B. R. Lickorish}, J. Knot Theory Ramifications 2, No. 2, 171-194 (1993; Zbl 0793.57003)].) Moreover, if \(n>2\) and \(K,L\) as above are links in \(S^3\) then \(V_K(A^{-4})=V_L(A^{-4})\), where \(V\) is the Jones polynomial. The author concludes that links in \(S^3\) with different Jones polynomials, for any choice of framings, can be \((n,2)\)-congruent for at most finitely many \(n\).
scientific article; zbMATH DE number 1008574