Singular Hecke algebras, Markov traces, and HOMFLY-type invariants

Paris, Luis; Rabenda, Loïc

Annales de l'Institut Fourier, Tome 58 (2008) no. 7, pp. 2413-2443.

On définit l’algèbre de Hecke singulière $\mathscr{H}\left(S{B}_n\right)$ comme le quotient de l’algèbre de monoïde $ℂ\left(q\right)\left[S{B}_n\right]$ par les relations de Hecke ${\sigma }_k^2=(q-1){\sigma }_k+q$, $1\le k\le n-1$. 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 ${\mathrm{TR}}_d$ l’ensemble des traces de Markov à $d$ points singuliers fixés. C’est un espace vectoriel sur $ℂ(q,z)$. Notre résultat principal est que ${\mathrm{TR}}_d$ est de dimension $d+1$. Ce résultat est complété par une construction explicite d’une base de ${\mathrm{TR}}_d$. 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 $\mathscr{H}\left(S{B}_n\right)$ as the quotient of the singular braid monoid algebra $ℂ\left(q\right)\left[S{B}_n\right]$ by the Hecke relations ${\sigma }_k^2=(q-1){\sigma }_k+q$, $1\le k\le n-1$. 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 ${\mathrm{TR}}_d$ denote the set of Markov traces with $d$ singular points. This is a $ℂ(q,z)$-vector space. Our main result is that ${\mathrm{TR}}_d$ is of dimension $d+1$. This result is completed with an explicit construction of a basis of ${\mathrm{TR}}_d$. 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.

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

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