Identifying the invariants for classical knots and links from the Yokonuma-Hecke algebras

Chlouveraki, Maria; Juyumaya, Jesús; Karvounis, Konstantinos; Lambropoulou, Sofia

doi:10.48550/arxiv.1505.06666

Cited by 10 publications

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“…The defining relations may also be expressed in terms of diagram. For instance, recall from [3] that relation (5) corresponds to move the tie from top to bottom behind or in front of the strand (see Figure13). For more details about type-A relations see [3, 5) and ( 6) in terms of diagrams (n = 3)…”

Section: The Tied Braid Monoid Of Type Bmentioning

confidence: 99%

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Tied links in the solid torus

Flores¹

2019

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Section: The Tied Braid Monoid Of Type Bmentioning

confidence: 99%

“…Subsequently, by using Jones' method invariants for: framed links [17], classical links [15] and singular links [16] were constructed. Moreover, recently it was proved that the invariants for classical links constructed in [15] are not topologically equivalent either to the Homflypt polynomial or to the Kauffman polynomial, see [5].…”

Section: Introductionmentioning

confidence: 99%

Tied links in the solid torus

Flores¹

2019

Preprint

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“…Yet they could be topologically equivalent to the Homflypt polynomial, in the sense that they might distinguish the same pairs of non-isotopic links. Eventually, in a recent development [2], another presentation for the Yokonuma-Hecke algebra is employed with parameter q in a new quadratic relation, where q 2 = u [6]. Using this presentation, the authors of [2] have been able to establish that the classical link invariants, Θ d , obtained from the isomorphic algebra Y d,n (q) coincide with the Homflypt polynomial on knots, but they are not topologically equivalent to the Homflypt polynomial on links (as it was conjectured in [7]).…”

Section: Introductionmentioning

confidence: 99%

“…Focusing now on the classical link invariants from the algebra FTL d,n (u), these need to be compared to the Jones polynomial. Following [2], in Section 7 we give a new presentation for the algebra FTL d,n with parameter q deriving from the new presentation of the Yokonuma-Hecke algebra Y d,n (q). We then adjust our results so far to the isomorphic algebra FTL d,n (q) and we apply them to the results of [2].…”

Section: Introductionmentioning

confidence: 99%

“…Following [2], in Section 7 we give a new presentation for the algebra FTL d,n with parameter q deriving from the new presentation of the Yokonuma-Hecke algebra Y d,n (q). We then adjust our results so far to the isomorphic algebra FTL d,n (q) and we apply them to the results of [2]. Namely, by specializing Θ d (q, z) to the our specific value for z, we obtain 1-variable invariants for classical knots and links, denoted by θ d (q).…”

Section: Introductionmentioning

confidence: 99%

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Framization of the Temperley–Lieb algebra

Goundaroulis

Juyumaya

Kontogeorgis

et al. 2017

Mathematical Research Letters

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Abstract. We propose a framization of the Temperley-Lieb algebra. The framization is a procedure that can briefly be described as the adding of framing to a known knot algebra in a way that is both algebraically consistent and topologically meaningful. Our framization of the Temperley-Lieb algebra is defined as a quotient of the Yokonuma-Hecke algebra. The main theorem provides necessary and sufficient conditions for the Markov trace defined on the Yokonuma-Hecke algebra to pass through to the quotient algebra. Using this we construct 1-variable invariants for classical knots and links, which, as we show, are not topologically equivalent to the Jones polynomial.

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A Survey on Temperley–Lieb-type Quotients from the Yokonuma–Hecke Algebras

Goundaroulis

2017

Springer Proceedings in Mathematics &Amp; Statistics

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In this survey we collect all results regarding the construction of the Framization of the Temperley-Lieb algebra of type A as a quotient algebra of the Yokonuma-Hecke algebra of type A. More precisely, we present all three possible quotient algebras the emerged during this construction and we discuss their dimension, linear bases, representation theory and the necessary and sufficient conditions for the unique Markov trace of the Yokonuma-Hecke algebra to factor through to each one of them. Further, we present the link invariants that are derived from each quotient algebra and we point out which quotient algebra provides the most natural definition for a framization of the Temperley-Lieb algebra. From the Framization of the Temperley-Lieb algebra we obtain new one-variable invariants for oriented classical links that, when compared to the Jones polynomial, they are not topologically equivalent since they distinguish more pair of non isotopic oriented links. Finally, we discuss the generalization of the newly obtained invariants to a new two-variable invariant for oriented classical links that is stronger than the Jones polynomial.

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Identifying the invariants for classical knots and links from the Yokonuma-Hecke algebras

Cited by 10 publications

References 0 publications

Tied links in the solid torus

Tied links in the solid torus

Framization of the Temperley–Lieb algebra

A Survey on Temperley–Lieb-type Quotients from the Yokonuma–Hecke Algebras

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