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Knot Polynomials from \(\mathcal{R}\)-Matrices: Wherefore This Mathematics?

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Abstract

This text is a continuation of the overview “Knot polynomials from \(\mathcal{R}\)-matrices: where is physics?” (Phys. Part. Nucl. 51, 172 (2020)). We continue to discuss the basics of a popular subject in modern mathematical physics: describing knots by means of \(\mathcal{R}\)-matrix polynomials. Having discussed the physical context, we now focus on the mathematical apparatus, in which topology, the theory of integrable systems, and the representation theory of quantum groups are intertwined. This text is intended as an introduction to the subject for all those interested in this topic.

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Notes

  1. A group representation is a set of linear operators (acting in the representation space, which is linear) corresponding to its elements so that the composition of operators corresponds to the product of elements; after choosing a basis in the representation space, the operators (representation elements) can be written in matrix form.

  2. The representation of an algebra differs from the representation of a group (see above) by an additional condition: a linear combination of elements of the algebra corresponds to the same linear combination of their matrices in the representation.

  3. This property is somewhat modified, depending on the chosen normalization of invariants [28].

  4. In the case of general position of parameters \(\lambda \) and \(\mu \) [32] (see Section 6.3.3).

  5. If both off-diagonal elements in a \(2 \times 2\) matrix \(A\) are nonzero, then changing the basis with the matrix \(V = {\text{diag}}(x,{{x}^{{ - 1}}})\), where \(x = \sqrt {A_{{2,1}}^{{ - 1}}{{A}_{{12}}}} \), will give a symmetric matrix \({{V}^{{ - 1}}}AV\).

  6. The correctness of this definition is discussed in Section 4.

  7. Note the difference in the definitions of direct and inverse twisted matrices: the first is obtained by permuting a pair of superscripts and the second is obtained by permuting a pair of subscripts. The reason for this agreement will be clarified in Section 5.2.

  8. Sequential writing of the equations for \(i = N,N - 1, \ldots ,1\) gives a linear system with a triangular matrix.

  9. With one essential supplement, to which Section 4.3.3 is devoted.

  10.  The author is grateful to Andrei Smirnov for pointing out this fact.

  11.  Each of the matrices \({{R}_{1}}\) and \({{R}_{2}}\) on such a subspace must have two different eigenvalues, otherwise the matrices commute and are diagonalized simultaneously.

  12.  See commentary in Section 6.1.3.

  13.  More precisely, all Lie algebras included in the Cartan classification [31].

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ACKNOWLEDGMENTS

I am grateful to A.Yu. Morozov and A.D. Mironov for many years of painstaking scientific guidance of my scientific work, and also to E.S. Suslova and V.V. Sleptsova (Nasonova) for creating conditions for it. I am grateful to E.T. Akhmedov, P.I. Dunin-Barkovskii, D.V. Vasil’ev, E.A. Vyrodov, D.V. Galakhov, A.A. Morozov, A.Yu. Orlov, I.V. Polyubin, A.V. Popolitov, A.A. Roslom, and A.V. Sleptsov for their interest in the work, careful reading of the drafts to this text, and numerous critical comments. I am also grateful to N.Ya. Amburg, G.B. Aminov, S.B. Artamonov, I.A. Danilenko, A.V. Zabrodin, O.S. Kruglinskaya, N.A. Nemkov, S.A. Mironov, Sh.R. Shakirov and other participants of the seminar of the Laboratory of Methods of Mathematical Physics at the Institute for Theoretical and Experimental Physics (ITEP) for regular fascinating and useful discussions. I am also grateful to I.V. Tyutin for a detailed analysis of the introductory example.

I am especially grateful to I.A. Dynnikov, A.V. Malyutin, and M.E. Kazaryan for their deep attention to the author’s work and valuable discussions.

I am grateful to E.T. Akhmedov, M.I. Vysotskii, A.V. Marshakov, V.A. Novikov, T.V. Uglov, and other organizers of the ITEP youth conference for the opportunity to speak to a new, diverse, and attentive audience.

I am also grateful to the people who opened my way to science, in particular, A.A. Abrikosov, D.A. Aleksandrov, M.V. Danilov, N.V.Z. Nozik, N. Ostrikov, V.P. Slobodyanin, T.V. Uglov, V.V. Shan’kov, and S.A. Sharakin.

Funding

This work was supported by the Russian Science Foundation, grant no. 16-12-10344.

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  1. Kurchatov Institute for Theoretical and Experimental Physics, National Research Center, 117218, Moscow, Russia

    A. S. Anokhina

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Anokhina, A.S. Knot Polynomials from \(\mathcal{R}\)-Matrices: Wherefore This Mathematics?. Phys. Part. Nuclei 52, 374–419 (2021). https://doi.org/10.1134/S1063779621030023

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