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 Ẑ 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.
Additional Information
© 2020 The Authors. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. Article funded by SCOAP3. Received 17 July 2020. Accepted 29 October 2020. Published 15 December 2020. We thank for discussions and acknowledge insightful comments from Tobias Ekholm, Stavros Garoufalidis, Angus Gruen, Sergei Gukov, Piotr Kucharski, Satoshi Nawata, Sunghyuk Park, Ramadevi Pichai, Vivek Singh and Marko Stošic. We also thank Robert Osburn for drawing our attention to the cyclotomic expansion of 8₁₉ knot. S.B. is supported by Humboldt postdoctoral grant. He also acknowledges his stay at the University of Warsaw where this project was initiated. J.J. was supported by the Polish National Science Centre (NCN) grant 2016/23/D/ST2/03125. The work of P.S. is supported by the TEAM programme of the Foundation for Polish Science co-financed by the European Union under the European Regional Development Fund (POIR.04.04.00-00-5C55/17-00).Attached Files
Published - Banerjee2020_Article_RevisitingTheMelvin-Morton-Roz.pdf
Accepted Version - 2007.00579
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Additional details
- Eprint ID
- 107171
- Resolver ID
- CaltechAUTHORS:20201217-143607089
- SCOAP3
- Alexander von Humboldt Foundation
- 2016/23/D/ST2/03125
- National Science Centre (Poland)
- Foundation for Polish Science
- POIR.04.04.00-00-5C55/17-00
- European Regional Development Fund
- Created
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2020-12-17Created from EPrint's datestamp field
- Updated
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2021-11-16Created from EPrint's last_modified field