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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With this we refer to the mechanism firstly conceived by V. Jones in  for the construction of the Homflypt polynomial.
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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
- Kauffman polynomial
- BMW algebra
- Jones polynomial
- Tied links
Mathematics Subject Classification