On the Kashaev signature conjecture (Q6622152)

For an oriented link diagram \( D \), Kashaev introduced a symmetric matrix indexed by the set of planar regions defined by the diagram [\textit{R. Kashaev}, IRMA Lect. Math. Theor. Phys. 33, 131--146 (2021; Zbl 1511.57005)]. He demonstrated that this matrix can be used to define an invariant of oriented links. Furthermore, he conjectured that both the Alexander polynomial \( \Delta_L(t) \) and the Levine-Tristram signature \( \sigma_L(w) \), where \( w \in S^1 \setminus \{1\} \), can be derived from this matrix.\N\NIn the paper under review, the authors confirm Kashaev's conjecture for the classical link signature \( \sigma_L(-1) \). Additionally, by connecting Kashaev's work to the Kauffman model for the Alexander polynomial [\textit{L. H. Kauffman}, Formal knot theory. Princeton University Press, Princeton, NJ (1983; Zbl 0537.57002)], they establish the part of the conjecture related to the Alexander polynomial. The authors further prove the full conjecture in the specific case of definite knots.