On Hecke lifting conjecture for framed knots

TL;DR

This work investigates the Hecke lifting conjecture for framed links motivated by the framed LMOV integrality framework in $U(N)$ Chern-Simons theory. It provides a direct proof of the conjecture for torus knots using explicit reformulated colored HOMFLY-PT invariants and a fractional twist map, and shows a limit-form verification for general framed knots, strengthening evidence for the conjecture. The paper also connects the conjecture to colored Alexander polynomials, showing that hook-partition colorings satisfy a precise limit relation and that the limit behavior aligns with known Alexander polynomial structure. The results illuminate the interplay between skein-theoretic reformulations, Adams operations, and integrality phenomena, offering concrete expressions and divisibility properties that support the framing-dependent LMOV program.

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.

Figures (4)

• Figure 1: $T_{d}^m$ is the closure of $(\beta_d)^m$
• Figure 2: Local relations
• Figure 3: Removal of an unknot
• Figure 4: $\mathcal{K}$ decorated by $\mathcal{Q}$

Theorems & Definitions (14)

• Conjecture 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 2.1
• Conjecture 2.2
• Lemma 3.1: cf. Lemma 4.2 in CLPZ23
• Proposition 3.2
• proof
• Lemma 3.3
• proof