The alternating torus-knot 3D index conjecture

Let p,q>0p,q>0 be coprime integers with pqeq0(mod2)pq eq 0\pmod{2}, and let T(p,q)\mathcal{T}_{(p,q)} be a 11-efficient ideal triangulation of the complement of the (p,q)(p,q) torus knot. The alternating torus-knot 3D index conjecture. The index satisfies

IT(p,q)(x,y)(q)=δ0,x+pqy.\mathcal{I}^{(x,y)}_{\mathcal{T}_{(p,q)}}(q)=\delta_{0,x+pqy}.

This generalizes the computed trefoil case and was conjectured on physical grounds for alternating torus knots. The supplied source does not indicate whether the conjecture has been resolved.

Sources & referencesView supporting material

Primary source

Daniele Celoria, Craig D. Hodgson and J. Hyam Rubinstein, “The 3D index and Dehn filling”, arXiv:2509.09886 (2025).

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