The t-deformed ADO polynomial limit conjecture for F_K

Let KK be a knot, let FK(x,a,q,t)F_K(x,a,q,t) be the (a,t)(a,t)-deformed FKF_K invariant, let ΔK(x,t)\Delta_K(x,t) be the tt-deformed Alexander polynomial, and let pp be a positive integer with ζp=e2πi/p\zeta_p=e^{2\pi i/p}. Define

ADOK(p;x,t):=ΔK(xp,(t)p)limqe2πi/pFK(x,t1q2,q,t).\operatorname{ADO}_K(p;x,t):=\Delta_K(x^p,-(-t)^p)\lim_{q\to e^{2\pi i/p}}F_K(x,-t^{-1}q^2,q,t).

The t-deformed ADO limit conjecture. The preceding expression should be a polynomial and, at t=1t=-1, should equal the usual pp-th ADO polynomial of KK for SU(2)SU(2). More generally, for positive integer NN, define

ADOKSU(N)(p;x,t):=ΔK(xp,(t)p)N1limqe2πi/pFK(x,t1qN,q,t).\operatorname{ADO}_K^{SU(N)}(p;x,t):=\Delta_K(x^p,-(-t)^p)^{N-1}\lim_{q\to e^{2\pi i/p}}F_K(x,-t^{-1}q^N,q,t).

This expression should be a polynomial and a tt-deformation of the symmetric version of the pp-th ADO polynomial for SU(N)SU(N). The conjecture extends the roots-of-unity relationship between FKF_K invariants and ADO polynomials, replacing the Alexander factor by its tt-deformed analogue. 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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