Revisiting the MelvinMortonRozansky expansion, or there and back again
Abstract
Alexander polynomial arises in the leading term of a semiclassical MelvinMortonRozansky expansion of colored knot polynomials. In this work, following the opposite direction, we propose how to reconstruct colored HOMFLYPT polynomials, superpolynomials, and newly introduced 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 MelvinMortonRozansky 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 HOMFLYPT homology and the knotsquivers correspondence.
 Publication:

Journal of High Energy Physics
 Pub Date:
 December 2020
 DOI:
 10.1007/JHEP12(2020)095
 arXiv:
 arXiv:2007.00579
 Bibcode:
 2020JHEP...12..095B
 Keywords:

 ChernSimons Theories;
 Topological Field Theories;
 Topological Strings;
 High Energy Physics  Theory;
 Mathematical Physics;
 Mathematics  Geometric Topology;
 Mathematics  Symplectic Geometry
 EPrint:
 Minor corrections, example of 8_19 knot added