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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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$U_q(sl(2))$-quantum invariants from an intersection of two Lagrangians in a symmetric power of a surface
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by Cristina Ana-Maria Anghel
Trans. Amer. Math. Soc. 376 (2023), 7139-7185
DOI: https://doi.org/10.1090/tran/8962
Published electronically: July 6, 2023

Abstract:

In this paper we show that coloured Jones and coloured Alexander polynomials can both be read off from the same picture provided by two Lagrangians in a symmetric power of a surface. More specifically, the $N^{th}$ coloured Jones and $N^{th}$ coloured Alexander polynomials are specialisations of a graded intersection between two explicit Lagrangian submanifolds in a symmetric power of the punctured disc. The graded intersection is parametrised by the intersection points between these Lagrangians, graded in a specific manner using the diagonals of the symmetric power. As a particular case, we see the original Jones and Alexander polynomials as two specialisations of a graded intersection between two Lagrangians in a configuration space, whose geometric supports are Heegaard diagrams.
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Bibliographic Information
  • Cristina Ana-Maria Anghel
  • Affiliation: University of Geneva, Section de mathématiques, Rue du Conseil-Général 7-9, Geneva CH 1205, Switzerland; and Institute of Mathematics “Simion Stoilow” of the Romanian Academy, 21 Calea Grivitei Street, 010702 Bucharest, Romania
  • MR Author ID: 1453438
  • Email: Cristina.Palmer-Anghel@unige.ch; cranghel@imar.ro
  • Received by editor(s): February 28, 2022
  • Received by editor(s) in revised form: November 26, 2022, March 10, 2023, and March 27, 2023
  • Published electronically: July 6, 2023
  • Additional Notes: The author was supported by SwissMAP, a National Centre of Competence in Research funded by the Swiss National Science Foundation as well as the Romanian Ministry of Education and Research, grant CNCS-UEFISCDI, project number PN-III-P4-ID-PCE-2020-2798.
  • © Copyright 2023 by the author
  • Journal: Trans. Amer. Math. Soc. 376 (2023), 7139-7185
  • MSC (2020): Primary 57K10; Secondary 57K14
  • DOI: https://doi.org/10.1090/tran/8962
  • MathSciNet review: 4636687