# zbMATH — the first resource for mathematics

Singular Hecke algebras, Markov traces, and HOMFLY-type invariants. (English) Zbl 1171.57008
Singular Hecke algebras $${\mathcal H}(SB_n)$$ are defined and Markov traces studied in this context, generalizing classical results. For example, it is proved that a Markov trace determines an invariant on links with $$d$$ singular points which satisfies a HOMFLY skein relation. The main result of the paper states that the dimension of the vector space of the Markov traces with $$d$$ singular points has dimension $$d+1$$. An explicit basis is constructed recursively: starting with the Ocneanu trace for $$d=0$$, the new traces are defined using two natural linear maps that preserve the number of strings of the braids, and reduce by one the number of singular points. This basis allows the authors to define a universal Markov trace and to obtain a universal HOMFLY-type invariant for singular links. This invariant is the unique invariant which satisfies some skein relation and some desingularization relation.
Two relevant open questions are also mentioned in the paper. The first one asks about the dimension of the Hecke algebra built with $$n$$ strings and $$d$$ singular points, for $$d\geq 1$$ and $$n\geq 3$$ (for $$d=0$$ it is $$n!$$, a basis given by the $$n!$$ positive permutation braids). Proposition 3.1 and Lemma 3.2 state that this dimension is strictly less than $$(n-1)^dn!$$. The second question asks if the natural morphism from $${\mathcal H}(SB_n)$$ to $$\mathcal H(SB_{n+1})$$ is injective.

##### MSC:
 57M25 Knots and links in the $$3$$-sphere (MSC2010) 20C08 Hecke algebras and their representations 20F36 Braid groups; Artin groups
Full Text:
##### References:
 [1] Baez, J. C., Link invariants of finite type and perturbation theory, Lett. Math. Phys., 26, 1, 43-51, (1992) · Zbl 0792.57002 [2] Birman, J. S., New points of view in knot theory, Bull. Amer. Math. Soc. (N.S.), 28, 2, 253-287, (1993) · Zbl 0785.57001 [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.), 12, 2, 239-246, (1985) · Zbl 0572.57002 [4] Garside, F. A., The braid group and other groups, Quart. J. Math. Oxford Ser., 20, 2, 235-254, (1969) · Zbl 0194.03303 [5] Gemein, B., Singular braids and markov’s theorem, J. Knot Theory Ramifications, 6, 4, 441-454, (1997) · Zbl 0885.57005 [6] Jones, V. F. R., A polynomial invariant for knots via von Neumann algebras, Bull. Amer. Math. Soc. (N.S.), 12, 1, 103-111, (1985) · Zbl 0564.57006 [7] Jones, V. F. R., Hecke algebra representations of braid groups and link polynomials, Ann. of Math., 126, 2, 335-388, (1987) · Zbl 0631.57005 [8] Kauffman, L. H., Invariants of graphs in three-space, Trans. Amer. Math. Soc., 311, 2, 697-710, (1989) · Zbl 0672.57008 [9] Kauffman, L. H.; Vogel, P., Link polynomials and a graphical calculus, J. Knot Theory Ramifications, 1, 1, 59-104, (1992) · Zbl 0795.57001 [10] Przytycki, J. H.; Traczyk, P., Invariants of links of Conway type, Kobe J. Math., 4, 2, 115-139, (1988) · Zbl 0655.57002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.