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AN INVARIANT FOR SINGULAR KNOTS

J. JUYUMAYA

Departamento de Matemáticas, Universidad de Valparaíso, Gran Bretaña 1091, Valparaíso, Chile

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and

S. LAMBROPOULOU

Department of Mathematics, National Technical University of Athens, Zografou Campus, GR-157 80 Athens, Greece

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https://doi.org/10.1142/S0218216509007324Cited by:37 (Source: Crossref)

Abstract

In this paper we introduce a Jones-type invariant for singular knots, using a Markov trace on the Yokonuma–Hecke algebras Y_d,n(u) and the theory of singular braids. The Yokonuma–Hecke algebras have a natural topological interpretation in the context of framed knots. Yet, we show that there is a homomorphism of the singular braid monoid SB_n into the algebra Y_d,n(u). Surprisingly, the trace does not normalize directly to yield a singular link invariant, so a condition must be imposed on the trace variables. Assuming this condition, the invariant satisfies a skein relation involving singular crossings, which arises from a quadratic relation in the algebra Y_d,n(u).

Keywords:

AMSC: 57M27, 20C08, 20F36

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