Detcherry's finiteness conjecture for link exteriors

Let R=Q(q12)R=Q(q^{\frac{1}{2}}), let LS3L\subset S^3 be a link with nn components, and let ELE_L denote its exterior. For each component, let tiS(EL)t_i\in\mathcal{S}(E_L) be the meridian of the ii-th component. Then R[t1,,tn]R[t_1,\ldots,t_n] is a subring of S(EL)\mathcal{S}(\partial E_L). Detcherry's finiteness conjecture. The skein module S(EL)\mathcal{S}(E_L) is finitely generated as an R[t1,,tn]R[t_1,\ldots,t_n]-module. This conjecture was proposed by Detcherry in 2021 and is disproved for the (3,3,3,3)(3,3,3,3)-pretzel link exterior, whose skein module is not finitely generated over Q(q12)[t1,t2]\mathbb{Q}(q^{\frac{1}{2}})[t_1,t_2].

Sources & referencesView supporting material

Primary source

Haimiao Chen, “Kauffman bracket skein module of the (3,3,3,3)-pretzel link exterior”, arXiv:2502.16234 (2025).

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