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

Luis Paris[1]; Loïc Rabenda[1]

  • [1] Université de Bourgogne Institut de Mathématiques de Bourgogne UMR 5584 du CNRS B.P. 47870 21078 Dijon cedex (France)

Annales de l’institut Fourier (2008)

  • Volume: 58, Issue: 7, page 2413-2443
  • ISSN: 0373-0956

Abstract

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We define the singular Hecke algebra ℋ ( S B n ) as the quotient of the singular braid monoid algebra ℂ ( q ) [ S B n ] by the Hecke relations σ k 2 = ( q - 1 ) σ k + q , 1 ≤ k ≤ 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 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 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.

How to cite

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Paris, Luis, and Rabenda, Loïc. "Singular Hecke algebras, Markov traces, and HOMFLY-type invariants." Annales de l’institut Fourier 58.7 (2008): 2413-2443. <http://eudml.org/doc/10383>.

@article{Paris2008,
abstract = {We define the singular Hecke algebra $\{\mathcal\{H\}\} (SB_n)$ as the quotient of the singular braid monoid algebra $\{\mathbb\{C\}\} (q) [SB_n]$ 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 $\{\rm TR\}_d$ denote the set of Markov traces with $d$ singular points. This is a $\{\mathbb\{C\}\} (q,z)$-vector space. Our main result is that $\{\rm TR\}_d$ is of dimension $d+1$. This result is completed with an explicit construction of a basis of $\{\rm TR\}_d$. 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.},
affiliation = {Université de Bourgogne Institut de Mathématiques de Bourgogne UMR 5584 du CNRS B.P. 47870 21078 Dijon cedex (France); Université de Bourgogne Institut de Mathématiques de Bourgogne UMR 5584 du CNRS B.P. 47870 21078 Dijon cedex (France)},
author = {Paris, Luis, Rabenda, Loïc},
journal = {Annales de l’institut Fourier},
keywords = {Singular Hecke algebra; singular link; singular knot; singular braid; Markov trace; singular Hecke algebra; Markov trace, HOMFLY-type invariant.},
language = {eng},
number = {7},
pages = {2413-2443},
publisher = {Association des Annales de l’institut Fourier},
title = {Singular Hecke algebras, Markov traces, and HOMFLY-type invariants},
url = {http://eudml.org/doc/10383},
volume = {58},
year = {2008},
}

TY - JOUR
AU - Paris, Luis
AU - Rabenda, Loïc
TI - Singular Hecke algebras, Markov traces, and HOMFLY-type invariants
JO - Annales de l’institut Fourier
PY - 2008
PB - Association des Annales de l’institut Fourier
VL - 58
IS - 7
SP - 2413
EP - 2443
AB - We define the singular Hecke algebra ${\mathcal{H}} (SB_n)$ as the quotient of the singular braid monoid algebra ${\mathbb{C}} (q) [SB_n]$ 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 ${\rm TR}_d$ denote the set of Markov traces with $d$ singular points. This is a ${\mathbb{C}} (q,z)$-vector space. Our main result is that ${\rm TR}_d$ is of dimension $d+1$. This result is completed with an explicit construction of a basis of ${\rm TR}_d$. 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.
LA - eng
KW - Singular Hecke algebra; singular link; singular knot; singular braid; Markov trace; singular Hecke algebra; Markov trace, HOMFLY-type invariant.
UR - http://eudml.org/doc/10383
ER -

References

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