A new proof of the faithfulness of the Temperley-Lieb representation of B3

• Autorzy: Piwocki A. H.
• Języki publikacji: EN

Abstrakty

EN

We find a way to calculate the term with lowest exponent in the Kauffman bracket polynomial of a closed 3-string braid. This leads us to a new proof of the faithfulness of the Temperley-Lieb representation of the 3-string braid group B3. In particular we will prove that for any link with braid index 3 either the coefficient of the lowest degree term or the coefficient of the highest degree term of the Jones polynomial is equal to ±1.

Słowa kluczowe

EN

Kauffman bracket polynomial; Jones polynomial; Temperley-Lieb algebra

PL

wielomian Kauffmana; wielomian Jonesa; algebra Temperley-Lieba

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2011
• Tom: Vol. 44, nr 2
• Strony: 433--443
• Opis fizyczny: Bibliogr. 11 poz., rys.

Twórcy

autor

Piwocki A. H.

• Department of Mathematics, Warsaw University, Banacha 2, 02–097 Warsaw, Poland

Bibliografia

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• [6] V. F. R.Jones, A polynomial invariant for knots via von Neumann algebras, Bull. Amer. Math. Soc. 12 (1985), 103–111.
• [7] V. F. R. Jones, The Jones polynomial, http://math.berkeley.edu/∼vfr/jones.pdf, (2005).
• [8] L. H. Kauffman, State models and the Jones polynomial, Topology 26 (1987), 395–407.
• [9] M. Kędziorek, Grupa warkoczy i jej dwie reprezentacje (eng. The Braid Group and two of its Representations), Bachelor dissertation, Department of Mathematics, Warsaw University (2008).
• [10] D. D. Long, M. Paton, The Burau representation is not faithful for n ≥ 6, Topology 32 (1993), 439–447.
• [11] J. A. Moody, The Burau representation of the braid group Bn is unfaithful for large n, Bull. Amer. Math. Soc. (N.S.) 25 (1991), 379–384.