Twisted torsion invariants and link concordance (Q2836443)

It is well known that the Alexander polynomial \(\Delta_K\) of a knot \(K\) may be interpreted as a Reidemeister torsion, and that if \(K\) is a slice knot then \(\Delta_K(t)=f(t)f(t^{-1})\), for some Laurent polynomial \(f\). These observations have been extended in various ways. This paper gives a further extension. Let \(L\) be an oriented \(m\)-component link, with group \(\pi\), and let \(\psi:\pi\to{H}\) be an epimorphism to a free abelian group, which maps each meridian non-trivially. Let \(R\subset\mathbb{C}\) be a subring which is closed under complex conjugation, and let \(Q(H)\) be the field of fractions of the group ring \(R[H]\). This has an involution \(f\mapsto\bar{f}\) which extends complex conjugation and the canonical involution of \(H\). Let \(\alpha:\pi\to{GL(R,k)}\) be a homorphism, with image a finite \(p\)-group, Then the twisted torsion \(\tau^{\alpha\otimes\psi}(L)\) is an element of \(Q(H)^\times/N\), where \(N\) is the subgroup of norms \(f\bar{f}\). It is shown that this is an invariant of link concordance, with a similar result for homology concordance of 3-manifolds.NEWLINENEWLINEThe argument involves two themes of relevance to link concordance, namely Poincaré duality and Stallings' theorem on the invariance of nilpotent quotients under homologically 2-connected maps of groups. The \(p\)-group assumption is also invoked in a key lemma which draws upon work of Strebel. The final section considers the case of satellite links and reproves a result of the first author, that the Bing double of the figure-eight knot is not slice.