The alternating torus-knot 3D index conjecture

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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.

References

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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