Revisiting the Melvin-Morton-Rozansky expansion, or there and back again

• Main Authors: Sibasish Banerjee, Jakub Jankowski, Piotr Sułkowski
• Format: Article
• Language: English
• Published: SpringerOpen 2020-12-01
• Series: Journal of High Energy Physics
• Subjects: Chern-Simons Theories; Topological Field Theories; Topological Strings
• Online Access: https://doi.org/10.1007/JHEP12(2020)095

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 Z ̂ $$ \hat{Z} $$ 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.