This article is cited in 3 scientific papers (total in 3 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,
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
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.
knot polynomial, superpolynomial, differential expansion.
|Russian Foundation for Basic Research
|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
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Theoretical and Mathematical Physics, 2017, 193:2, 1630–1646
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
\by Ya.~A.~Kononov, A.~Yu.~Morozov
\paper Rectangular superpolynomials for the~figure-eight knot $4_1$
\jour Theoret. and Math. Phys.
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Morozov A., “Homfly For Twist Knots and Exclusive Racah Matrices Inrepresentation ”, Phys. Lett. B, 778 (2018), 426–434
Morozov A., “Knot Polynomials For Twist Satellites”, Phys. Lett. B, 782 (2018), 104–111
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