Knot-quiver skein relation conjecture

From papers

Let KK be a knot with corresponding quiver QKQ_K. For each crossing of type L+L_+, let Ki,τK_{i,\tau}, with τ()\tau\binom{}{} replaced by τeq\tau eq? be the knot or link obtained by changing or smoothing the ii-th crossing, where τewcommand\tau ewcommand{}{} is one of the resolutions - and 00. The corresponding quivers are denoted QKi,τQ_{K_{i,\tau}}. Knot-quiver skein relation conjecture. For every resolution i,τi,\tau, QKi,τQ_{K_{i,\tau}} is a subquiver of QKQ_K in an appropriate framing, and QKQ_K admits a block form composed from any two such resolutions. This conjectural lifting of the HOMFLY-PT skein relation would encode all symmetric-coloured HOMFLY-PT skein relations at the quiver level; its status is not resolved in the supplied text.

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Primary source

Sachin Chauhan, Piotr Kucharski, Dmitry Noshchenko, Ramadevi Pichai, Vivek Kumar Singh and Marko Stošić, “Full twists and stability of knots and quivers”, arXiv:2508.18417 (2025).

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