HOMFLY-PT $a$-deformed Gukov–Manolescu conjecture

For every knot $K\subset S^3$, let $F_K^N(x,q)=F_K^{\mathfrak{sl}_N,\mathrm{sym}}(x,q)$ denote the series associated with symmetric representations, let $\Delta_K(x)$ be the Alexander polynomial, let $P_k(K;a,q)$ denote the coloured HOMFLY-PT polynomials, and let $\hat A_K(\hat x,\hat y,a,q)$ be the $a$-deformed quantum A-polynomial. HOMFLY-PT $a$-deformation conjecture. There exists a three-variable series $F_K(x,a,q)$ such that

$$F_K(x,q^N,q)=F_K^N(x,q),\qquad \hat A_K(\hat x,\hat y,a,q)F_K(x,a,q)=0.$$

Moreover,

$$F_K(x,a,q)=F_K(a^{-1}x^{-1},a,q),\qquad F_K(x,1,q)=\Delta_K(x),$$ $$F_K(x,q,q)=1,\qquad \lim_{q\to1}F_K(x,q^N,q)=\frac{1}{\Delta_K(x)^{N-1}},$$

and its asymptotic expansion satisfies

$$\log F_K(e^{k\hbar},a,e^{\hbar})=\log P_k(K;a,e^{\hbar})$$

as a series in $\hbar$. This conjecture seeks a HOMFLY-PT analogue interpolating all symmetric $\mathfrak{sl}_N$ series and incorporating quantum A-polynomial, Weyl-symmetry, and classical-limit constraints. The source presents it as an unproved conjecture and gives computational evidence for selected knots.