The (a,t)-deformed F_K interpolation conjecture for knots

Let KS3K\subset S^3 be a knot. Let FK(x,a,q)F_K(x,a,q) be the three-variable function from the aa-deformed FKF_K conjecture, let FK(x,a,q,t)F_K(x,a,q,t) be a four-variable refinement, let A^K(x^,y^,a,q,t)\hat{A}_K(\hat{x},\hat{y},a,q,t) be the quantum super-AA-polynomial, and let Pr(K;a,q,t)\mathcal{P}_r(K;a,q,t) be the coloured superpolynomial.

The (a,t)-deformed FKF_K conjecture. There should exist a four-variable function satisfying

FK(x,a,q,1)=FK(x,a,q),A^K(x^,y^,a,q,t)FK(x,a,q,t)=0,F_K(x,a,q,-1)=F_K(x,a,q),\qquad \hat{A}_K(\hat{x},\hat{y},a,q,t)F_K(x,a,q,t)=0,

and, as an \hbar-series,

logFK(er,a,e,t)=logPr(K;a,e,t).\log F_K(e^{r\hbar},a,e^\hbar,t)=\log\mathcal{P}_r(K;a,e^\hbar,t).

This conjecture seeks a tt-deformation of the preceding FKF_K invariant that specializes at t=1t=-1 and captures superpolynomial asymptotics. The supplied source gives no resolution status.

Sources & referencesView supporting material

Primary source

Tobias Ekholm, Angus Gruen, Sergei Gukov, Piotr Kucharski, Sunghyuk Park and Piotr Sułkowski, “Z at large N: from curve counts to quantum modularity”, arXiv:2005.13349 (2020).

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