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1995 | 34 | 1 | 121-148

Tytuł artykułu

Search for different links with the same Jones' type polynomials: Ideas from graph theory and statistical mechanics

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Warianty tytułu

Języki publikacji

EN

Abstrakty

EN
We describe in this talk three methods of constructing different links with the same Jones type invariant. All three can be thought as generalizations of mutation. The first combines the satellite construction with mutation. The second uses the notion of rotant, taken from the graph theory, the third, invented by Jones, transplants into knot theory the idea of the Yang-Baxter equation with the spectral parameter (idea employed by Baxter in the theory of solvable models in statistical mechanics). We extend the Jones result and relate it to Traczyk's work on rotors of links. We also show further applications of the Jones idea, e.g. to 3-string links in the solid torus. We stress the fact that ideas coming from various areas of mathematics (and theoretical physics) has been fruitfully used in knot theory, and vice versa.

Słowa kluczowe

Rocznik

Tom

34

Numer

1

Strony

121-148

Opis fizyczny

Daty

wydano
1995

Twórcy

  • Department of Mathematics, University of California, Berkeley, CA 94720, USA

Bibliografia

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  • [3] R. Baxter, Exactly solved models in statistical mechanics, Academic Press, London, 1982.
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  • [8] J. Hoste, A polynomial invariant of knots and links, Pacific J. Math. 124 (1986), 295-320.
  • [9] J. Hoste, J. H. Przytycki, A survey of skein modules of 3-manifolds, in: Knots 90, De Gruyter, Berlin - New York 1992, 363-379.
  • [10] J. Hoste, J. H. Przytycki, Tangle surgeries which preserve Jones-type polynomials, Center for Pure and Applied Mathematics preprint - PAM 617, U. C. Berkeley, 1994.
  • [11] G. T. Jin, D. Rolfsen, Some remarks on rotors in link theory, Canad. Math. Bull. 34 (1991), 480-484.
  • [12] V. Jones, Hecke algebra representations of braid groups and link polynomials, Ann. of Math. 126 (1987), 335-388.
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  • [15] V. F. R. Jones, Commuting transfer matrices and link polynomials, Internat. J. Math. 3 (1992), 205-212.
  • [16] V. F. R. Jones, Coincident link polynomials from commuting transfer matrices, Proceedings of the XXth International Conference on Differential Geometric Methods in Theoretical Physics, Vol. 1, 2 (New York, 1991), 137-151, World Sci. Publishing, River Edge, NJ, 1992.
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  • [18] T. Kanenobu, The Homfly and the Kauffman bracket polynomials for the generalized mutant of a link, Topology Appl., to appear.
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  • [26] M-T-1 H. R. Morton, P. Traczyk, The Jones polynomial of satellite links around mutants, in: Braids, Ed. J. S. Birman, A. Libgober, AMS Contemporary Math., 78 (1988), 587-592.
  • [27] H. R. Morton, P. Traczyk, Knots and algebras, Contribuciones Matematicas en homenaje al profesor D. Antonio Plans Sanz de Bremond, ed. E. Martin-Peinador and A. Rodez Usan, University of Zaragoza, (1990), 201-220.
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  • [32] J. H. Przytycki, Applications of the spectral parameter tangle of V. Jones, Abstracts Amer. Math. Soc. 12 (1991), 496-497.
  • [33] J. H. Przytycki, The spectral parameter 3-string tangle, in preparation.
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  • [36] D. Rolfsen, The quest for a knot with trivial Jones polynomial; diagram surgery and the Temperley-Lieb algebra, in: Topics in knot theory, Ed. M. E. Bozhüyük, NATO ASI Series, Series C: Mathematical and Physical Sciences - Vol. 399, Kluwer Academic Publishers 1993, 195-210.
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  • [38] H. N. V. Temperley, E. H. Lieb, Relations between the 'percolation' and 'coloring' problem and other graph-theoretical problems associated with regular planar lattices: some exact results for the 'percolation' problem, Proc. Roy. Soc. London Ser. A 322 (1971), 251-280.
  • [39] P. Traczyk, A note on rotant links, preprint, 1989.
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  • [41] V. G. Turaev, The Conway and Kauffman modules of the solid torus, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 167 (1988), 79-89. %English translation: J. Soviet Math.
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