Generalized knots–quivers correspondence

Establish that, for any knot K, there exists a symmetric quiver Q with m vertices and a specialization of the quiver variables of the form x_i = (-1)^{s_i} a^{a_i} q^{q_i} x^{n_i}, where n_i are nonnegative integers, such that the quiver generating series P_C(x_1, …, x_m) equals the generating function P(K)(x,a,q) of appropriately normalized symmetrically colored HOMFLY‑PT polynomials of K.

Background

The original knots–quivers correspondence relates the generating series of colored HOMFLY‑PT polynomials for a knot to the generating series of a symmetric quiver, after a linear specialization of quiver variables. Motivated by recent developments and the super‑exponential growth of colored HOMFLY‑PT polynomials for larger knots, the paper proposes allowing higher powers of the series variable x in the specialization x_i = (−1)^{s_i} a^{a_i} q^{q_i} x^{n_i}.

This generalization aims to capture broader classes of knots, including homologically thick knots and those with super‑exponential behavior, while maintaining the quiver framework that encodes colored HOMFLY‑PT invariants. The conjecture asserts the existence of such a quiver and specialization for every knot.