Revisiting the Melvin-Morton-Rozansky Expansion, or There and Back Again

Abstract

Alexander polynomial arises in the leading term of a semi-classical Melvin-Morton-Rozansky expansion of colored knot polynomials. In this work, following the opposite direction, we propose how to reconstruct colored HOMFLY-PT polynomials, superpolynomials, and newly introduced $\hat{Z}$ invariants for some knot complements, from an appropriate rewriting, quantization and deformation of Alexander polynomial. Along this route we rederive conjectural expressions for the above mentioned invariants for various knots obtained recently, thereby proving their consistency with the Melvin-Morton-Rozansky theorem, and derive new formulae for colored superpolynomials unknown before. For a given knot, depending on certain choices, our reconstruction leads to equivalent expressions, which are either cyclotomic, or encode certain features of HOMFLY-PT homology and the knots-quivers correspondence.

Minor corrections, example of 8_19 knot added

References in corpus (5)

• Large N Duality, Mirror Symmetry, and a Q-deformed A-polynomial for Knots
• BPS states, knots and quivers
• $\hat{Z}$ at large $N$: from curve counts to quantum modularity
• Multi-cover skeins, quivers, and 3d $\mathcal{N}=2$ dualities
• An analytic version of the Melvin-Morton-Rozansky Conjecture

Cited by in corpus (3)

• Branches, quivers, and ideals for knot complements
• On knots, complements, and 6j-symbols
• Quantum Curve for strip geometries, Topological Recursion and open GW/DT invariants