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Original knots–quivers correspondence

Establish that, for any knot K, there exists a quiver Q such that the quiver generating series P_C(x_1, …, x_m) matches the generating series P(K)(x,a,q) of the colored HOMFLY‑PT polynomials of K after the linear variable specializations x_i = (−1)^{s_i} a^{a_i} q^{q_i} x for i = 1,…,m.

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Background

The paper recalls the original (non‑generalized) knots–quivers correspondence, which postulates that colored HOMFLY‑PT generating series can be represented by a quiver generating series under linear specializations of the quiver variables. This sets the baseline for the generalized version introduced later, and clarifies the exact matching of generating functions via specified substitutions for x_i.

The authors restate the conjectural nature of this correspondence and provide formal definitions of the quiver form of colored HOMFLY‑PT polynomials to make the framework precise.

References

Knots-quivers correspondence was originally postulated in . For a given knot $K$ it conjectures the existence of a quiver $Q$ such that the corresponding quiver generating series (\ref{P-C}) matches the generating series (\ref{kser}) of the colored HOMFLY-PT of the knot $K$, after specializations: \begin{equation}\label{xx}x_i=(-1){s_i}a{a_i}q{q_i}x,\quad i=1,\ldots,m,\end{equation} for suitable integers $a_i,$ $q_i$, and $s_i$, for $i=1,\ldots,m$.

xx:

xi=(−1)siaaiqqix,i=1,…,m,x_i=(-1)^{s_i}a^{a_i}q^{q_i}x,\quad i=1,\ldots,m,

— Generalized knots-quivers correspondence (2402.03066 - Stošić, 5 Feb 2024) in Section “Notation” (after equations (P-C) and (kser))