The Links–Gould (LG) polynomials are two-variable link invariants arising from the quantum supergroup $U_q(\mathfrak{sl}(2|1))$, exhibiting hybrid behavior between the Jones and Alexander polynomials.  I will survey some recent results illustrating this theme.

The basic LG invariant provides a lower bound on the Seifert genus and this bound extends to its colored versions.  It has been verified by Garoufalidis and Li that the 2-colored Links–Gould polynomial detects the genus for all 352.2 million prime knots with up to 19 crossings.  We also prove a conjecture of Geer and Patureau-Mirand that the Links–Gould invariant admits a specialization to the Akutsu–Deguchi–Ohtsuki (ADO) invariant at a sixth root of unity.  Finally, we establish analogs of Jones–Wenzl idempotents for LG and formulate an analog of the Fox conjecture, supported by computational evidence.

This is joint work, in various combinations, with Stavros Garoufalidis, Rinat Kashaev, Ben-Michael Kohli, Jiebo Song, Guillaume Tahar, and Emmanuel Wagner.