2d log-concavity conjecture for Links–Gould coefficients of alternating links

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Let LL be an alternating link, and write

LG(L;t0,t1)=∑i,jaijt0it1j.\mathrm{LG}(L;t_0,t_1)=\sum_{i,j}a_{ij}t_0^i t_1^j.

A two-indexed sequence (bij)(b_{ij}) is 2d log-concave if, for every i,j,k,l∈Zi,j,k,l\in\mathbb Z,

bi+k,j+lbi−k,j−l≤bij2.b_{i+k,j+l}b_{i-k,j-l}\leq b_{ij}^2.

2d log-concavity conjecture. The absolute values (∣aij∣)(|a_{ij}|) form a 2d log-concave sequence with no interior zeros. This conjecture is proposed as a bidimensional analogue of log-concavity for Alexander-polynomial coefficients; the supplied source gives no resolution.

References

Primary source

Matthew Harper, Ben-Michael Kohli, Jiebo Song and Guillaume Tahar, “On some log-concavity properties of the Alexander-Conway and Links-Gould invariants”, arXiv:2509.16868 (2025).

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