Ishii’s alternating-sign conjecture for the Links–Gould polynomial of alternating knots

Prove that for every alternating knot K, the coefficients a_{ij} of the Links–Gould polynomial LG(K; t0, t1) = Σ_{i,j} a_{ij} t0^i t1^j satisfy the alternating sign property: a_{ij} a_{i'j'} ≥ 0 whenever i + j − i' − j' is even, and a_{ij} a_{i'j'} ≤ 0 otherwise.

Background

The Alexander polynomial of an alternating link is known to be alternating, and Ishii proposed an analogous property for the two-variable Links–Gould invariant. The paper cites Ishii’s conjecture and records computational evidence verified up to 16 crossings via computations related to the Garoufalidis–Kashaev V1-polynomial. The conjecture asserts a precise parity-controlled sign pattern for all coefficients of LG for alternating knots.

References

In his early foundational studies on the Links-Gould invariant, Ishii conjectured that $LG$ displays the same ``alternating'' behavior as the Alexander polynomial on alternating knots.

— On some log-concavity properties of the Alexander-Conway and Links-Gould invariants (2509.16868 - Harper et al., 21 Sep 2025) in Section 1.3, immediately before Conjecture [Ishii]