On rack cohomology

Etingof, Pavel; Graña, M.

doi:10.1016/s0022-4049(02)00159-7

Cited by 65 publications

(77 citation statements)

References 5 publications

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“…We first review the rack complexes introduced by [13] (see also [10]) and the quandle homologies defined in [3]. For a quandle X , we denote by C R n (X ) the free Z[As(X )]-module generated by n-elements of X .…”

Section: Reviews Of Rack Homology Quandle Homology and Topological Mmentioning

confidence: 99%

Quandle homotopy invariants of knotted surfaces

Nosaka

2012

Math. Z.

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Given a finite quandle, we introduce a quandle homotopy invariant of knotted surfaces in the 4-sphere, modifying that of classical links. This invariant is valued in the third homotopy group of the quandle space, and is universal among the (generalized) quandle cocycle invariants. We compute the second and third homotopy groups, with respect to "regular Alexander quandles". As a corollary, any quandle cocycle invariant using the dihedral quandle of prime order is a scalar multiple of Mochizuki 3-cocycle invariant. As another result, we determine the third quandle homology group of the dihedral quandle of odd order.Keywords Quandle · Classical link · Knotted surface · Rack space · Quandle cocycle invariant · k-invariant · Homotopy group · Loop space IntroductionA quandle is an algebraic system satisfying axioms that correspond to the Reidemeister moves. Given a quandle X , Fenn et al. [13] defined the rack space B X, in analogy to the classifying spaces of groups; Further, an invariant of framed links in S 3 was proposed in [14, §4] which is called a quandle homotopy invariant and is valued in the second homotopy group π 2 (B X) [see [23] for some computations of π 2 (B X)]. In addition, as a modification of the homology H * (B X; Z), Carter et al. [4] introduced its quandle homology denoted by H Q n (X ; A), and further quandle cocycle invariants of classical links (resp. linked surfaces) using cohomology classes in H 2 Q (X ; A) (resp. H 3 Q (X ; A)). For its applications, the homology groups H * (B X; A) and H Q * (X ; A) of some quandles X have been computed [5,8,19,20,24,26]. Furthermore the quandle cocycle invariants were generalized to allow the cohomology H * Q (X ; A) with local coefficients [3]. T. Nosaka (B)Research Institute for Mathematical Sciences, Kyoto University, Sakyo-ku, Kyoto 606-8502, Japan e-mail: nosaka@kurims.kyoto-u.ac.jp 123 342 T. NosakaAs for classical links, the quandle cocycle invariants were much studied (see, e.g., [15,23,25,31]), and are known to be derived from the quandle homotopy invariant above (see [6,31]). The study of the homotopy group π 2 (B X) was useful to understood a topological meaning of some quandle cocycle invariants [15].In this paper, we introduce and study a quandle homotopy invariant of oriented linked surfaces valued in a group ring Z[π Q 3 (B X)] (Definition 2.3), modifying the above quandle homotopy invariant of links in S 3 . Here the group π Q 3 (B X) is defined by a certain quotient group of the third homotopy group π 3 (B X) (see Remark 2.2 for details). Similar to the quandle homotopy invariant of classical links, that of linked surfaces is shown to be universal among the quandle cocycle invariants with local coefficients of linked surfaces (see Sect. 2.3). It is therefore significant to estimate and determine π Q 3 (B X). Moreover, from the study of π Q 3 (B X), we address a problem of determining those local coefficients for which the the associated quandle cocycle invariants pick out completely the quandle homotopy invariant.First, we de...

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Section: Reviews Of Rack Homology Quandle Homology and Topological Mmentioning

confidence: 99%

Quandle homotopy invariants of knotted surfaces

Nosaka

2012

Math. Z.

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“…P. Etingof and M. Graña [11] have calculated rack cohomology H * (Q, A) assuming | Inn(Q)| invertible in A. Our calculation of H 2 YB (c Q , A) generalizes their result from diagonal to general Yang-Baxter deformations.…”

Section: Related Workmentioning

confidence: 56%

Yang–Baxter deformations of quandles and racks

Eisermann

2005

Algebr. Geom. Topol.

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Given a rack Q and a ring A, one can construct a Yang-Baxter operator c Q :y for all x, y ∈ Q. In answer to a question initiated by D.N. Yetter and P.J. Freyd, this article classifies formal deformations of c Q in the space of Yang-Baxter operators. For the trivial rack, where x y = x for all x, y , one has, of course, the classical setting of r-matrices and quantum groups. In the general case we introduce and calculate the cohomology theory that classifies infinitesimal deformations of c Q . In many cases this allows us to conclude that c Q is rigid. In the remaining cases, where infinitesimal deformations are possible, we show that higher-order obstructions are the same as in the quantum case.

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“…Remark 1.4. Yang-Baxter cohomology includes rack cohomology H * R (Q; Λ) as its diagonal part, as explained in §3, where Λ is a module over some ring K. If | Inn(Q)| is invertible in K, then H * R (Q; Λ) is trivial in a certain sense [17]. In the modular case, however, it leads to non-trivial and topologically interesting Yang-Baxter deformations (see Example 7.7 below).…”

Section: Notation (Racks)mentioning

confidence: 99%

Yang–Baxter deformations and rack cohomology

Eisermann¹

2014

Trans. Amer. Math. Soc.

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ABSTRACT. Every rack Q provides a set-theoretic solution c Q of the Yang-Baxter equation by setting c Q : x ⊗ y → y ⊗ x y for all x,y ∈ Q. This article examines the deformation theory of c Q within the space of Yang-Baxter operators over a ring A, a problem initiated by Freyd and Yetter in 1989. As our main result we classify deformations in the modular case, which had previously been left in suspense, and establish that every deformation of c Q is gauge-equivalent to a quasi-diagonal one. Stated informally, in a quasi-diagonal deformation only behaviourally equivalent elements interact. In the extreme case, where all elements of Q are behaviourally distinct, Yang-Baxter cohomology thus collapses to its diagonal part, which we identify with rack cohomology. The latter has been intensively studied in recent years and, in the modular case, is known to produce non-trivial and topologically interesting Yang-Baxter deformations.

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On rack cohomology

Cited by 65 publications

References 5 publications

Quandle homotopy invariants of knotted surfaces

Quandle homotopy invariants of knotted surfaces

Yang–Baxter deformations of quandles and racks

Yang–Baxter deformations and rack cohomology

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