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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
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