Direct proof of cut-edge independence for the determinant formula

Establish a direct proof that the determinant expression for the Links–Gould polynomial is independent of the choice of cut edge in a connected link diagram, without relying on the partial-trace properties (P_1) and (P_2) of the R-matrix.

Background

For a connected link diagram, the paper expresses the Links–Gould polynomial using two edge-indexed determinants with the row and column corresponding to a chosen cut edge deleted. The resulting product is proved to be independent of that edge by invoking the partial-trace properties (P_1) and (P_2) of the R-matrix.

The authors explicitly state that they do not know a direct proof of this independence, making the derivation of such a proof an unresolved methodological problem.

References

We do not know a direct proof of this independence: here it comes from Theorem~\ref{thm.GHKSTp}, that is, from the properties (P$_1$) and (P$_2$) of the $R$-matrix.

— The Links--Gould polynomial of an alternating link is alternating  (2610.18803 - Harper et al., 13 Oct 2026) in Remark 6.3(i), Section 6.3, “The edge formula for an arbitrary diagram”