The Jones and Alexander Polynomials for Singular Links

Fiedler, Thomas

doi:10.1142/s0218216510008236

Cited by 16 publications

(11 citation statements)

References 12 publications

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“…Together with the introduction of this extension, the notion of finite type (or Vassiliev) invariants, as invariants vanishing on some step of this filtration, was introduced producing a new point of view on knot theory. Since then, many knot invariants, as well as different knot representations and techniques have been extended to singular knots and links (see for example [Bir93,Fie10]). Recently, in [CEHN17], a singular link invariant having the form of a binary algebraic structure and called singquandle, was defined; as the name suggests, this structure extends to the singular case the quandle invariant for classical links.…”

Section: Introductionmentioning

confidence: 99%

On the axioms of singquandles

Bonatto,

Cattabriga

2021

Preprint

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In this paper we deal with the notion of singquandles introduced in [CEHN17]. This is an algebraic structure that naturally axiomatizes Reidemeister moves for singular links, similarly to what happens for ordinary links and quandle structure. We present a new axiomatization that shows different algebraic aspects and simplifies applications. We also reformulate and simplify the axioms for affine singquandles (in particular in the idempotent case).

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Section: Introductionmentioning

confidence: 99%

On the axioms of singquandles

Bonatto,

Cattabriga

2021

Preprint

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“…A proof of this conjecture was given by Paris in [27]. Kauffman state models of the Alexander and Jones polynomials were investigated by Fiedler in the context of singular knots [17]. Quandle theory was extended to the context of non-oriented singular knots in [13] in order to provide invariants for singular knots.…”

Section: Introductionmentioning

confidence: 99%

Singquandles, Psyquandles and Singular Knots: A Survey

Ceniceros¹,

Churchill²,

Elhamdadi³

et al. 2021

Preprint

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In this short survey we review recent results dealing with algebraic structures (quandles, psyquandles, and singquandles) related to singular knot theory. We first explore the singquandles counting invariant and then consider several recent enhancements to this invariant. These enhancements include a singquandle cocycle invariant and several polynomial invariants of singular knots obtained from the singquandle structure. We then explore psyquandles which can be thought of as generalizations of oriented signquandles, and review recent developments regarding invariants of singular knots obtained from psyquandles.

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“…Note that if we ignore the A-regions, then these moves are similar to the moves governing the isotopy of singular knots given by Kauffman in [3]. For more on singular knots and their invariants see [4][5][6].…”

Section: Introductionmentioning

confidence: 99%

Stuck Knots

Bataineh

2020

Symmetry

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Singular knots and links have projections involving some usual crossings and some four-valent rigid vertices. Such vertices are symmetric in the sense that no strand overpasses the other. In this research we introduce stuck knots and links to represent physical knots and links with projections involving some stuck crossings, where the physical strands get stuck together showing which strand overpasses the other at a stuck crossing. We introduce the basic elements of the theory and we give some isotopy invariants of such knots including invariants which capture the chirality (mirror imaging) of such objects. We also introduce another natural class of stuck knots, which we call relatively stuck knots, where each stuck crossing has a stuckness factor that indicates to the value of stuckness at that crossing. Amazingly, a generalized version of Jones polynomial makes an invariant of such quantized knots and links. We give applications of stuck knots and links and their invariants in modeling and understanding bonded RNA foldings, and we explore the topology of such objects with invariants involving multiplicities at the bonds. Other perspectives are also discussed.

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The Jones and Alexander Polynomials for Singular Links

Cited by 16 publications

References 12 publications

On the axioms of singquandles

On the axioms of singquandles

Singquandles, Psyquandles and Singular Knots: A Survey

Stuck Knots

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