Singular Hecke algebras, Markov traces, and HOMFLY-type invariants
[Algèbres de Hecke singulières, traces de Markov et invariants de type HOMFLY]
Annales de l'Institut Fourier, Tome 58 (2008) no. 7, pp. 2413-2443.

On définit l’algèbre de Hecke singulière (SB n ) comme le quotient de l’algèbre de monoïde (q)[SB n ] par les relations de Hecke σ k 2 =(q-1)σ k +q, 1kn-1. On définit la notion de trace de Markov dans ce cadre, en fixant le nombre d de points singuliers, et on démontre qu’une trace de Markov détermine un invariant sur les entrelacs à d points singuliers qui vérifie une relation d’écheveau. Soit TR d l’ensemble des traces de Markov à d points singuliers fixés. C’est un espace vectoriel sur (q,z). Notre résultat principal est que TR d est de dimension d+1. Ce résultat est complété par une construction explicite d’une base de TR d . Grâce à ces résultats, nous définissons une trace de Markov universelle et un invariant universel de type HOMFLY sur les entrelacs singuliers. Cet invariant est l’unique invariant qui vérifie une certaine relation d’écheveau et une certaine relation de désingularisation.

We define the singular Hecke algebra (SB n ) as the quotient of the singular braid monoid algebra (q)[SB n ] by the Hecke relations σ k 2 =(q-1)σ k +q, 1kn-1. We define the notion of Markov trace in this context, fixing the number d of singular points, and we prove that a Markov trace determines an invariant on the links with d singular points which satisfies some skein relation. Let TR d denote the set of Markov traces with d singular points. This is a (q,z)-vector space. Our main result is that TR d is of dimension d+1. This result is completed with an explicit construction of a basis of TR d . Thanks to this result, we define a universal Markov trace and a universal HOMFLY-type invariant on singular links. This invariant is the unique invariant which satisfies some skein relation and some desingularization relation.

DOI : 10.5802/aif.2419
Classification : 57M25, 20C08, 20F36
Keywords: Singular Hecke algebra, singular link, singular knot, singular braid, Markov trace
Mot clés : Algèbre de Hecke singulière, entrelacs singulier, noeud singulier, trace de Markov
Paris, Luis 1 ; Rabenda, Loïc 1

1 Université de Bourgogne Institut de Mathématiques de Bourgogne UMR 5584 du CNRS B.P. 47870 21078 Dijon cedex (France)
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Paris, Luis; Rabenda, Loïc. Singular Hecke algebras, Markov traces, and HOMFLY-type invariants. Annales de l'Institut Fourier, Tome 58 (2008) no. 7, pp. 2413-2443. doi : 10.5802/aif.2419. http://www.numdam.org/articles/10.5802/aif.2419/

[1] Baez, J. C. Link invariants of finite type and perturbation theory, Lett. Math. Phys., Volume 26 (1992) no. 1, pp. 43-51 | DOI | MR | Zbl

[2] Birman, J. S. New points of view in knot theory, Bull. Amer. Math. Soc. (N.S.), Volume 28 (1993) no. 2, pp. 253-287 | DOI | MR | Zbl

[3] Freyd, P.; Yetter, D.; Hoste, J.; Lickorish, W. B. R.; Millett, K.; Ocneanu, A. A new polynomial invariant of knots and links, Bull. Amer. Math. Soc. (N.S.), Volume 12 (1985) no. 2, pp. 239-246 | DOI | MR | Zbl

[4] Garside, F. A. The braid group and other groups, Quart. J. Math. Oxford Ser., Volume 20 (1969) no. 2, pp. 235-254 | DOI | MR | Zbl

[5] Gemein, B. Singular braids and Markov’s theorem, J. Knot Theory Ramifications, Volume 6 (1997) no. 4, pp. 441-454 | DOI | Zbl

[6] Jones, V. F. R. A polynomial invariant for knots via von Neumann algebras, Bull. Amer. Math. Soc. (N.S.), Volume 12 (1985) no. 1, pp. 103-111 | DOI | MR | Zbl

[7] Jones, V. F. R. Hecke algebra representations of braid groups and link polynomials, Ann. of Math., Volume 126 (1987) no. 2, pp. 335-388 | DOI | MR | Zbl

[8] Kauffman, L. H. Invariants of graphs in three-space, Trans. Amer. Math. Soc., Volume 311 (1989) no. 2, pp. 697-710 | DOI | MR | Zbl

[9] Kauffman, L. H.; Vogel, P. Link polynomials and a graphical calculus, J. Knot Theory Ramifications, Volume 1 (1992) no. 1, pp. 59-104 | DOI | MR | Zbl

[10] Przytycki, J. H.; Traczyk, P. Invariants of links of Conway type, Kobe J. Math., Volume 4 (1988) no. 2, pp. 115-139 | MR | Zbl

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