Twisted quandle homology theory and cocycle knot invariants

Carter, J Scott; Elhamdadi, Mohamed; Saito, Masahico

doi:10.2140/agt.2002.2.95

Cited by 49 publications

(46 citation statements)

References 24 publications

(53 reference statements)

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“…If the module A is a homogeneous Alexander module as defined in example 2.4, then extensions of X by A are exactly the twisted quandle extensions described by Carter, Saito and Elhamdadi [6], and so Ext Q (X, A) = H 2 T Q (X; A).…”

Section: Abelian Extensions Of Quandlesmentioning

confidence: 99%

“…We then develop an Abelian extension theory for racks and quandles which contains the variants developed by Carter, Elhamdadi, Kamada and Saito [6,7] as special cases.…”

Section: Introductionmentioning

confidence: 99%

“…We further show that this definition coincides with the appropriate specialisation of the definition developed by Beck [3], and hence that these objects form a suitable category of coefficient objects in which to develop homology and cohomology theories for racks and quandles.We then develop an Abelian extension theory for racks and quandles which contains the variants developed by Carter, Elhamdadi, Kamada and Saito [6,7] as special cases. …”

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Extensions of racks and quandles

Jackson

2005

Homology, Homotopy and Applications

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A rack is a set equipped with a bijective, self-right-distributive binary operation, and a quandle is a rack which satisfies an idempotency condition.In this paper, we introduce a new definition of modules over a rack or quandle, and show that this definition includes the one studied by Etingof and Graña [9] and the more general one given by Andruskiewitsch and Graña [1]. We further show that this definition coincides with the appropriate specialisation of the definition developed by Beck [3], and hence that these objects form a suitable category of coefficient objects in which to develop homology and cohomology theories for racks and quandles.We then develop an Abelian extension theory for racks and quandles which contains the variants developed by Carter, Elhamdadi, Kamada and Saito [6,7] as special cases.

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Section: Abelian Extensions Of Quandlesmentioning

confidence: 99%

“…We then develop an Abelian extension theory for racks and quandles which contains the variants developed by Carter, Elhamdadi, Kamada and Saito [6,7] as special cases.…”

Section: Introductionmentioning

confidence: 99%

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confidence: 99%

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Extensions of racks and quandles

Jackson

2005

Homology, Homotopy and Applications

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“…Many variations of this idea exist, including the quandle homology described in [3], the twisted quandle homology described in [4], the quandle homology with coefficients in a quandle module ( [1], [10]), the biquandle version known as YangBaxter homology ( [5]), and a unifying approach to the homology of quandles and Lie algebras ( [2]). …”

Section: Yang-baxter Cohomologymentioning

confidence: 99%

Virtual Yang-Baxter cocycle invariants

Ceniceros

Nelson

2009

Trans. Amer. Math. Soc.

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Abstract. We extend the Yang-Baxter cocycle invariants for virtual knots by augmenting Yang-Baxter 2-cocycles with cocycles from a cohomology theory associated to a virtual biquandle structure. These invariants coincide with the classical Yang-Baxter cocycle invariants for classical knots but provide extra information about virtual knots and links. In particular, they provide a method for detecting non-classicality of virtual knots and links.

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“…Owing to the difficulty of computing the fundamental quandle of a link, it is natural to try to obtain partial information by developing a convenient homology theory, viewed as a way to define sort of linear approximations. Initiated by R. Fenn, C. Rourke, B. Sanderson from 1990 [43,45] and developed by S. Carter, M. Elhamdadi, M. Saito, and their collaborators in [14,15,16,17], this approach proved to be extremely fruitful, as explained in [43,18].…”

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confidence: 99%

Laver’s results and low-dimensional topology

Dehornoy

2015

Arch. Math. Logic

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Abstract. In connection with his interest in selfdistributive algebra, Richard Laver established two deep results with potential applications in low-dimensional topology, namely the existence of what is now known as the Laver tables and the well-foundedness of the standard ordering of positive braids. Here we present these results and discuss the way they could be used in topological applications.Richard Laver established two remarkable results that might lead to significant applications in low-dimensional topology, namely the existence of a series of finite structures satisfying the left-selfdistributive law, now known as the Laver tables, and the well-foundedness of the standard ordering of Artin's positive braids. In this text, we shall explain the precise meaning of these results and discuss their (past or future) applications in topology. In one word, the current situation is that, although the depth of Laver's results is not questionable, few topological applications have been found. However, the example of braid groups orderability shows that, once initial obstructions are solved, topological applications of algebraic results involving selfdistributivity can be found; the situation with Laver tables is presumably similar, and the only reason explaining why so few applications are known is that no serious attempt has been made so far, mainly because the results themselves remain widely unknown in the topology community.Therefore this text is more a program than a report on existing results. Our aim is to provide a self-contained and accessible introduction to the subject, hopefully helping the algebraic and topological communities to better communicate. Most of the results mentioned below have already appeared in literature, a number of them even belonging to the folklore of their domain (whereas ignored outside of it). However, at least the observations about cocycles for Laver tables mentioned in Subsection 1.3 (and established in another paper) are new.Very naturally, the text comprises two sections, one devoted to Laver tables, and one devoted to the well-foundedness of the braid ordering. It should be noted that the above two topics (Laver tables, well-foundedness of the braid ordering) do not exhaust Laver's contributions to selfdistributive algebra and, from there, to potential topological applications: in particular, Laver constructed powerful tools for investigating free LD-structures, leading to applications of their own [68,69,70]. However, connections with topology are less obvious in these cases and we shall not develop them here (see the other articles in this volume).

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Twisted quandle homology theory and cocycle knot invariants

Cited by 49 publications

References 24 publications

Extensions of racks and quandles

Extensions of racks and quandles

Virtual Yang-Baxter cocycle invariants

Laver’s results and low-dimensional topology

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