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Multi-colored Links From 3-Strand Braids Carrying Arbitrary Symmetric Representations

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Abstract

Obtaining HOMFLY-PT polynomials \(H_{R_1,\ldots ,R_l}\) for arbitrary links with l components colored by arbitrary SU(N) representations \(R_1,\ldots ,R_l\) is a very complicated problem. For a class of rank r symmetric representations, the [r]-colored HOMFLY-PT polynomial \(H_{[r_1],\ldots ,[r_l]}\) evaluation becomes simpler, but the general answer lies far beyond our current capabilities. To simplify the situation even more, one can consider links that can be realized as a 3-strand closed braid. Recently (Itoyama et al. in Int J Mod Phys A28:1340009, 2013. arXiv:1209.6304), it was shown that \(H_{[r]}\) for knots realized by 3-strand braids can be constructed using the quantum Racah coefficients (6j-symbols) of \(U_q(sl_2)\), which makes easy not only to evaluate such invariants, but also to construct analytical formulas for \(H_{[r]}\) of various families of 3-strand knots. In this paper, we generalize this approach to links whose components carry arbitrary symmetric representations. We illustrate the technique by evaluating multi-colored link polynomials \(H_{[r_1],[r_2]}\) for the two-component link L7a3 whose components carry \([r_1]\) and \([r_2]\) colors. Using our results for exclusive Racah matrices, it is possible to calculate symmetric-colored HOMFLY-PT polynomials of links for the so-called one-looped links, which are obtained from arborescent links by adding a loop. This is a huge class of links that contains the entire Rolfsen table, all 3-strand links, all arborescent links, and, for example, all mutant knots with 11 intersections.

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Notes

  1. The usual framing factor in front of quantum trace, which provide the invariance under the first Reidemeister move, we incorporate in \({\mathcal {R}}\)-matrix by modifying its eigenvalues (10).

  2. Recall, that if \(Q_{\nu }\) is a Young diagram \(Q_\nu =\{Q_{\nu _1}\ge Q_{\nu _2}\ge \cdots Q_{\nu _l}>0\}\), then the highest weights \(\mathbf {\omega }_{Q_{\nu }}\) of the corresponding representation are \(\omega _i= Q_{\nu _i}-Q_{\nu _{i+1}} \ \forall \, i=1,\dots ,l\), and vice versa \(Q_{\nu _i} = \sum _{k=i}^l \, \omega _k\).

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Acknowledgements

Our work was partly supported by the Grant of the Foundation for the Advancement of Theoretical Physics “BASIS” (A.M’s and A.S.), by Grant 16-31-60082-mol-a-dk (A.S.) and 18-31-20046-mol-a-ved (A.S.), by RFBR Grants 16-01-00291 (A.Mir.), 16-02-01021 (A.Mor.) and 17-01-00585 (An.Mor.), by joint Grants 17-51-50051-YaF, 18-51-05015-Arm-a (A.M’s, A.S.), 18-51-45010-Ind-a (A.M’s, A.S.), 19-51-53014-Gfen-a (A.M’s, A.S.), 19-51-50008-Yaf-a (And.Mir., And.Mor.) and by President of Russian Federation Grant MK-2038.2019.1 (And.Mor.). PR, VKS and SD acknowledge DST-RFBR Grant (INT/RUS/RFBR/P-309) for support. SD would like to thank CSIR for research fellowship. VKS would like to acknowledge the ERC Starting Grant no. 335739 “Quantum fields and knot homologies” for support. A. Mir., A. Mor., An. Mor. and PR would like to thank Kavli Institute for Theoretical Physics at the University of California, Santa Barbara, for local hospitality where we discussed some parts of this work.

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Authors and Affiliations

  1. Department of Physics, Indian Institute of Technology Bombay, Mumbai, 400076, India

    Saswati Dhara & Pichai Ramadevi

  2. Lebedev Physics Institute, Moscow, Russia, 119991

    Andrei Mironov

  3. ITEP, Moscow, Russia, 117218

    Andrei Mironov, Alexei Morozov, Andrey Morozov & Alexey Sleptsov

  4. Institute for Information Transmission Problems, Moscow, Russia, 127994

    Andrei Mironov, Alexei Morozov, Andrey Morozov & Alexey Sleptsov

  5. Moscow Institute of Physics and Technology, Dolgoprudny, Russia, 141701

    Andrey Morozov & Alexey Sleptsov

  6. Faculty of Physics, University of Warsaw, ul. Pasteura 5, 02-093, Warsaw, Poland

    Vivek Kumar Singh

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  1. Saswati Dhara
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Dhara, S., Mironov, A., Morozov, A. et al. Multi-colored Links From 3-Strand Braids Carrying Arbitrary Symmetric Representations. Ann. Henri Poincaré 20, 4033–4054 (2019). https://doi.org/10.1007/s00023-019-00841-z

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