  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.
