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.
Highlights
By Rozansky to more general knot polynomials [2,3,4]
It is believed that a general knot homology theory exists, which encompasses both knot Floer homology — which categorifies Alexander polynomial — and a putative HOMFLY-PT homology [7, 8]
The main goal of this note is to understand in detail how the Melvin-Morton-Rozansky expansion arises from various recently found expressions for colored knot polynomials, and — taking the opposite direction — to provide prescription how to reconstruct colored HOMFLY-PT polynomials and superpolynomials starting from Alexander polynomial
Summary
Alexander polynomial ∆(x) can be defined in various ways: by considering a knot diagram and associating weights to its crossings, in terms of a Seifert matrix, by Conway’s skein relations, etc. This provides a t-deformation of Alexander polynomial, and can be obtained as a specialization of a superpolynomial, i.e. Poincaré characteristic of HOMFLY-PT homology, briefly discussed
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