The L2-Alexander invariant is stronger than the genus and the simplicial volume

Abstract

We study how the genus, the simplicial volume and the [Formula: see text]-Alexander invariant of Li and Zhang can detect individual knots among all others. In particular, we use various techniques coming from hyperbolic geometry and topology to prove that the [Formula: see text]-Alexander invariant contains strictly more information than the pair (genus, simplicial volume). Along the way, we prove that the [Formula: see text]-Alexander invariant detects the figure-eight knot [Formula: see text], the twist knot [Formula: see text] and an infinite family of cables on the figure-eight knot.