A classifying invariant of knots, the knot quandle

doi:10.1016/0022-4049(82)90077-9

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A classifying invariant of knots, the knot quandle

Journal Article•DOI•

A classifying invariant of knots, the knot quandle

David Joyce¹•Institutions (1)

Clark University¹

01 Jan 1982-Journal of Pure and Applied Algebra (North-Holland)-Vol. 23, Iss: 1, pp 37-65

TL;DR: The Wirtinger presentation of the knot group involves only relations of the form y -1 xy = z and so may be construed as presenting a quandle rather than a group as discussed by the authors.

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About: This article is published in Journal of Pure and Applied Algebra.The article was published on 1982-01-01 and is currently open access. It has received 999 citations till now. The article focuses on the topics: Knot group & Knot invariant.

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Journal Article•DOI•

Finite Type Invariants of Classical and Virtual Knots

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Mikhail Goussarov, Michael Polyak¹, Oleg Viro²•Institutions (2)

Tel Aviv University¹, Uppsala University²

01 Sep 2000-Topology

TL;DR: In this paper, the isotopy classification problem for virtual knots is reduced to an algebraic problem formulated in terms of an algebra of arrow diagrams, and a new notion of finite type invariant is introduced.

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549 citations

Journal Article•DOI•

Quandle cohomology and state-sum invariants of knotted curves and surfaces

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J. Scott Carter¹, Daniel Jelsovsky², Daniel Jelsovsky³, Seiichi Kamada⁴, Seiichi Kamada⁵, Laurel Langford⁶, Masahico Saito² - Show less +3 more•Institutions (6)

University of South Alabama¹, University of South Florida², Florida Southern College³, Osaka City University⁴, Hiroshima University⁵, University of Wisconsin-Madison⁶

24 Jun 2003-Transactions of the American Mathematical Society

TL;DR: The 2-twist trefoil is an example of a sphere that is knotted in 4-dimensional space as mentioned in this paper, and a proof is given in this paper that this sphere is distinct from the same sphere with its orientation reversed.

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Abstract: The 2-twist spun trefoil is an example of a sphere that is knotted in 4-dimensional space. A proof is given in this paper that this sphere is distinct from the same sphere with its orientation reversed. Our proof is based on a state-sum invariant for knotted surfaces developed via a cohomology theory of racks and quandles (also known as distributive groupoids). A quandle is a set with a binary operation - the axioms of which model the Reidemeister moves in classical knot theory. Colorings of diagrams of knotted curves and surfaces by quandle elements, together with cocycles of quandles, are used to define state-sum invariants for knotted circles in 3-space and knotted surfaces in 4-space. Cohomology groups of various quandles are computed herein and applied to the study of the state-sum invariants. Non-triviality of the invariants is proved for a variety of knots and links, and conversely, knot invariants are used to prove non-triviality of cohomology for a variety of quandles.

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413 citations

Book Chapter•DOI•

An Introduction to Knot Theory

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Louis H. Kauffman¹•Institutions (1)

University of Illinois at Chicago¹

01 Jan 2001

TL;DR: In this paper, the construction of invariants of knots, such as the Jones polynomials and the Vassiliev invariants, and the relationships of these invariants to other mathematics (such as Lie algebras) are investigated.

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Abstract: This paper concentrates on the construction of invariants of knots, such as the Jones polynomials and the Vassiliev invariants, and the relationships of these invariants to other mathematics (such as Lie algebras).

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410 citations

Journal Article•DOI•

From racks to pointed Hopf algebras

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Nicolás Andruskiewitsch¹, Matías Graña²•Institutions (2)

National University of Cordoba¹, Massachusetts Institute of Technology²

25 Sep 2003-Advances in Mathematics

TL;DR: In this article, the authors studied the structure of racks, their cohomology groups and the corresponding Nichols algebras of braided vector spaces arising from groups, and showed advances in these three directions.

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359 citations

Cites background from "A classifying invariant of knots, t..."

...s called the defect group of X in the literature and coincides with the subgroup Γ considered in [So, Th. 2.6]. The name “enveloping group” is justiﬁed by the following fact, contained essentially in [J1]: Lemma 1.6. The functor X → GX is left adjoint to the forgetful functor H → FH from the category of groups to that of racks. That is, Homgroups(GX,H) ≃ Homracks(X,FH) by natural isomorphisms. The deﬁ...
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...−1)x; it is called a homogeneous crossed set. FROM RACKS TO POINTED HOPF ALGEBRAS 11 It can be shown that a crossed set Xis homogeneous if and only if it is a single orbit under the action of Aut⊲(X) [J1]. Twisted homogeneous crossed sets. In the same vein, let Gbe a group and s∈ Aut(G). Take x⊲y= xs(yx−1). This is a crossed set which is diﬀerent, in general, to the previous one. Any orbit of this is ...
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... = φiφjφ −1 i = φiyφ −1 i = φi⊲y; the claim follows. Similarly, i−1 ⊲F y⊆ Fφ−1 i ⊲y , hence (1). Now (2) follows from (1). We give ﬁnally some deﬁnitions of special classes of crossed sets, following [J1]. Deﬁnition 1.22. Let (X,⊲) be a quandle. We shall say that Xis involutory if φ2 x = id for all x∈ X. That is, if x⊲(x⊲y) = yfor all x,y∈ X. We shall say that Xis abelian if (x⊲w)⊲(y⊲z) = (x⊲y)⊲(w⊲z) ...
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...: Im(f) + xi → Im(f), τi(y) = y− xi are isomorphisms. FROM RACKS TO POINTED HOPF ALGEBRAS 13 Involutory crossed sets. This is a discretization of symmetric spaces, which we shall roughly present (see [J1] for the full explanation). Let Sbe a set provided with a collection of functions γ : Z → S, called geodesics, such that any two points of S belong to the image of some of them. Consider the aﬃne cros...
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...Swith the structure of an involutory crossed set if it maps geodesics to geodesics; that is, if x⊲γis a geodesic for any xand any γ. It can be shown that any involutory crossed set arises in this way [J1]. Core crossed sets. Let Gbe a group. The core of Gis the crossed set (G,⊲), where x⊲y= xy−1x. The core is an involutory crossed set. If Gis abelian, its core is the aﬃne crossed set with g= −id. More...
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Monograph•DOI•

3-Manifold Groups

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Matthias Aschenbrenner, Stefan Friedl, Henry Wilton

20 Aug 2015-arXiv: Geometric Topology

TL;DR: In this paper, the authors summarize properties of 3-manifold groups, with a particular focus on the consequences of the recent results of Ian Agol, Jeremy Kahn, Vladimir Markovic and Dani Wise.

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Abstract: We summarize properties of 3-manifold groups, with a particular focus on the consequences of the recent results of Ian Agol, Jeremy Kahn, Vladimir Markovic and Dani Wise.

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336 citations