Kauffman type invariants for tied links

Abstract

We define two new invariants for tied links. One of them can be thought as an extension of the Kauffman polynomial and the other one as an extension of the Jones polynomial which is constructed via a bracket polynomial for tied links. These invariants are more powerful than both the Kauffman and the bracket polynomials when evaluated on classical links. Further, the extension of the Kauffman polynomial is more powerful of the Homflypt polynomial, as well as of certain new invariants introduced recently. Also we propose a new algebra which plays in the case of tied links the same role as the BMW algebra for the Kauffman polynomial in the classical case. Moreover, we prove that the Markov trace on this new algebra can be recovered from the extension of the Kauffman polynomial defined here.

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Change history

Notes

  1. 1.

    With this we refer to the mechanism firstly conceived by V. Jones in [9] for the construction of the Homflypt polynomial.

References

  1. 1.

    Aicardi, F., Juyumaya, J.: An algebra involving braids and ties, ICTP Preprint IC/2000/179, see also arxiv:1709.03740

  2. 2.

    Aicardi, F., Juyumaya, J.: Markov trace on the algebra of braid and ties. Moscow Math. J. 16(3), 397–431 (2016)

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Aicardi, F., Juyumaya, J.: Tied Links. J. Knot theory ramifications. 25(9), 28 (2016). https://doi.org/10.1142/S02182165164100171

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Aicardi, F.: New invariants of links from a skein invariant of colored links, see arXiv:1512.00686

  5. 5.

    Birman, J.S., Wenzl, H.: Braids, links polynomials and a new algebra. Trans. Amer. Math. Soc. 313(1), 249–273 (1989)

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Cha, J. C., Livingston, C.: LinkInfo: Table of Knot Invariants, http://www.indiana.edu/~linkinfo (2015). Accessed 16 Apr 2015

  7. 7.

    Cohen, A.M., et al.: BMW algebras of simply laced type. J. Algebra 286, 107–153 (2005)

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Chlouveraki et al., M.: Identifying the invariants for classical knots and links from the Yokonuma–Hecke algebras. http://arxiv.org/pdf/1505.06666

  9. 9.

    Jones, V.F.R.: Hecke algebra representations of braid groups and link polynomials. Ann. Math. 126, 335–388 (1987)

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Jones, V.F.R.: Subfactors and knots, CBMS regional conference series in mathematics, vol. 80, p. 113. American Mathematical Society, Providence, RI (1991)

  11. 11.

    Lickorish, W.B.R.: An introduction to knot theory, graduate texts in mathematics 175. Springer, New York (1991)

    Google Scholar 

  12. 12.

    Morton, H., Wassermann, A.: A basis for the Birman–Wenzl algebra, unpublished manuscript, 1989, revised (2000)

  13. 13.

    Murakami, J.: The Kauffman polynomial of links and representation theory. Osaka J. Math. 24, 745–758 (1987)

    MathSciNet  MATH  Google Scholar 

  14. 14.

    Kauffman, L.: States model and the Jones polynomial. Topology 26(3), 395–407 (1987)

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    Kauffman, L.: An invariant of regular isotopy. Trans. Am. Math. Soc. 318(2), 417–471 (1990)

    MathSciNet  Article  MATH  Google Scholar 

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Correspondence to Francesca Aicardi.

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The authors has been supported partially by Fondecyt 1141254.

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Aicardi, F., Juyumaya, J. Kauffman type invariants for tied links. Math. Z. 289, 567–591 (2018). https://doi.org/10.1007/s00209-017-1966-0

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Keywords

  • Kauffman polynomial
  • BMW algebra
  • Jones polynomial
  • Tied links

Mathematics Subject Classification

  • 57M25
  • 20C08
  • 20F36