The $L^2$-Alexander invariant is stronger than the genus and the simplicial volume

TL;DR

This paper demonstrates that the $L^2$-Alexander invariant provides more detailed information than traditional invariants like genus and simplicial volume, enabling the detection of specific knots such as the figure-eight knot and certain cables.

Contribution

It proves that the $L^2$-Alexander invariant is strictly stronger than the pair (genus, simplicial volume) in knot detection, using hyperbolic geometry and topology techniques.

Findings

$L^2$-Alexander invariant detects the figure-eight knot $4_1$

It detects the twist knot $5_2$

It identifies an infinite family of cables on the figure-eight knot

Abstract

We study how the genus, the simplicial volume and the $L^2$-Alexander invariant of W. Li and W. Zhang can detect individual knots among all others. In particular, we use various techniques coming from hyperbolic geometry and topology to prove that the $L^2$-Alexander invariant contains strictly more information than the pair (genus, simplicial volume). Along the way we prove that the $L^2$-Alexander invariant detects the figure-eight knot $4_1$, the twist knot $5_2$ and an infinite family of cables on the figure-eight knot.