On Hecke lifting conjecture for framed knots

Abstract

Motivated by an amazing integrality structure conjecture for the $U(N)$ Chern-Simons quantum invariants of framed knots investigated by Mariño and Vafa, a new conjectural formula, named Hecke lifting conjecture, was proposed in \cite{CLPZ23} for framed links. This note is devoted to the study of this Hecke lifting conjecture. We prove this conjecture for torus knots using the explicit formulas of colored HOMFLY-PT invariants of torus knots, and we also verify the conjecture in a limit form for any framed knots.