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TMF, 2017, Volume 193, Number 2, Pages 256–275 (Mi tmf9327)  

This article is cited in 5 scientific papers (total in 5 papers)

Rectangular superpolynomials for the figure-eight knot $4_1$

Ya. A. Kononovab, A. Yu. Morozovcde

a National Research University "Higher School of Economics", Moscow, Russia
b Landau Institute for Theoretical Physics of Russian Academy of Sciences, Chernogolovka, Moscow Oblast, Russia
c Institute for Theoretical and Experimental Physics, Moscow, Russia
d National Research Nuclear University MEPhI (Moscow Engineering Physics Institute), Moscow, Russia
e Institute for Information Transmission Problems of the Russian Academy of Sciences, Moscow, Russia

Abstract: We rewrite the recently proposed differential expansion formula for HOMFLY polynomials of the knot $4_1$ in an arbitrary rectangular representation $R=[r^s]$ as a sum over all Young subdiagrams $\lambda$ of $R$ with surprisingly simple coefficients of the $Z$ factors. Intriguingly, these coefficients are constructed from the quantum dimensions of symmetric representations of the groups $SL(r)$ and $SL(s)$ and restrict the summation to diagrams with no more than $s$ rows and $r$ columns. Moreover, the $\beta$-deformation to Macdonald dimensions yields polynomials with positive integer coefficients, which are plausible candidates for the role of superpolynomials for rectangular representations. Both the polynomiality and the positivity of the coefficients are nonobvious, nevertheless true. This generalizes the previously known formulas for symmetric representations to arbitrary rectangular representations. The differential expansion allows introducing additional gradings. For the trefoil knot $3_1$, to which our results for the knot $4_1$ are immediately extended, we obtain the so-called fourth grading of hyperpolynomials. The property of factorization in roots of unity is preserved even in the five-graded case.

Keywords: knot polynomial, superpolynomial, differential expansion.

Funding Agency Grant Number
Russian Foundation for Basic Research 15-51-52031-HHC_а
15-52-50041-YaF
16-51-53034-GFEN
16-51-45029-Ind
16-01-00291
15-31-20832-мол_a_вед
16-02-01021
16-31-00484-мол_а
Simons Foundation
This research is supported by the Russian Foundation for Basic Research (Grant Nos. 15-51-52031-HHC_a, 15-52-50041-YaF, 16-51-53034-GFEN, and 16-51-45029-Ind).
The research of Ya. A. Kononov is supported in part by the Russian Foundation for Basic Research (Grant Nos. 16-01-00291 and 16-31-00484-mol_a) and the Simons Foundation.
The research of A. Yu. Morozov is supported in part by the Russian Foundation for Basic Research (Grant Nos. 16-02-01021 and 15-31-20832-mol_a_ved).


DOI: https://doi.org/10.4213/tmf9327

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English version:
Theoretical and Mathematical Physics, 2017, 193:2, 1630–1646

Bibliographic databases:

Received: 20.12.2016

Citation: Ya. A. Kononov, A. Yu. Morozov, “Rectangular superpolynomials for the figure-eight knot $4_1$”, TMF, 193:2 (2017), 256–275; Theoret. and Math. Phys., 193:2 (2017), 1630–1646

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. Morozov, “Generalized hypergeometric series for Racah matrices in rectangular representations”, Mod. Phys. Lett. A, 33:4 (2018), 1850020  crossref  mathscinet  zmath  isi  scopus
    2. A. Morozov, “Homfly for twist knots and exclusive Racah matrices inrepresentation [333]”, Phys. Lett. B, 778 (2018), 426–434  crossref  mathscinet  zmath  isi  scopus
    3. A. Morozov, “Knot polynomials for twist satellites”, Phys. Lett. B, 782 (2018), 104–111  crossref  mathscinet  isi  scopus
    4. Morozov A., “On Exclusive Racah Matrices (S)Over-Bar For Rectangular Representations”, Phys. Lett. B, 793 (2019), 116–125  crossref  isi
    5. Morozov A., “Extension of Kntz Trick to Non-Rectangular Representations”, Phys. Lett. B, 793 (2019), 464–468  crossref  isi
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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