On concordances in 3-manifolds

© 2018 London Mathematical Society We describe an action of the concordance group of knots in S 3 on concordances of knots in arbitrary 3-manifolds. As an application we define the notion of almost-concordance between knots. After some basic results, we prove the existence of non-trivial almost-concordance classes in all non-abelian 3-manifolds. Afterwards, we focus the attention on the case of lens spaces, and use a modified version of the Ozsváth–Szabó–Rasmussen's τ-invariant to obstruct almost-concordances and prove that each L(p, 1) admits infinitely many nullhomologous non almost-concordant knots. Finally we prove an inequality involving the cobordism PL-genus of a knot and its τ-invariants, in the spirit of [Sarkar, Math. Res. Lett. 18 (2011) 1239–1257].