Geometric Interpretations of Quandle Homology

Carter, J. Scott; Kamada, Seiichi; Saito, Masahico

doi:10.1142/s0218216501000901

Cited by 107 publications

(125 citation statements)

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“…We now reformulate the (generalized) quandle cocycle invariants in [3,4,6] from our quandle homotopy invariant. We first remark that, if B X Y is path-connected, the inclusion (1) has no 1-cell by definition.…”

Section: Reconstrucion Of Generalized Quandle Cocycle Invariantsmentioning

confidence: 99%

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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: Reconstrucion Of Generalized Quandle Cocycle Invariantsmentioning

confidence: 99%

“…As 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].…”

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confidence: 99%

Quandle homotopy invariants of knotted surfaces

Nosaka

2012

Math. Z.

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“…On the other hand, knot diagrams colored by quandles can be used to study quandle homology groups. This viewpoint was developed in [15,16,19] for rack homology and homotopy, and generalized to quandle homology in [8].…”

Section: Introductionmentioning

confidence: 99%

Twisted quandle homology theory and cocycle knot invariants

Carter

Elhamdadi

Saito

2002

Algebr. Geom. Topol.

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The quandle homology theory is generalized to the case when the coefficient groups admit the structure of Alexander quandles, by including an action of the infinite cyclic group in the boundary operator. Theories of Alexander extensions of quandles in relation to low dimensional cocycles are developed in parallel to group extension theories for group cocycles. Explicit formulas for cocycles corresponding to extensions are given, and used to prove non-triviality of cohomology groups for some quandles. The corresponding generalization of the quandle cocycle knot invariants is given, by using the Alexander numbering of regions in the definition of statesums. The invariants are used to derive information on twisted cohomology groups.

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“…As is known [RS,N1,CKS], given a quandle 2-cocycle φ : X 2 → A, the pairing between this φ and the cycle invariant Φ X (L) coincides with the original cocycle invariant in [CJKLS,Theorem 4.4]. Although this Φ X (L) is constructed from link diagrams, in §5 we later explain its topological meaning, together with a computation of H Q 2 (X) following from Eisermann [E1, E2].…”

Section: Review; Quandle Homotopy Invariant Of Linksmentioning

confidence: 98%

Homotopical interpretation of link invariants from finite quandles

Nosaka

2015

Topology and its Applications

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This paper demonstrates a topological meaning of quandle cocycle invariants of links with respect to finite connected quandles X, from a perspective of homotopy theory: Specifically, for any prime ℓ which does not divide the type of X, the ℓ-torsion of this invariants is equal to a sum of the colouring polynomial and a Z-equivariant part of the Dijkgraaf-Witten invariant of a cyclic branched covering space. Moreover, our homotopical approach involves application of computing some third homology groups and second homotopy groups of the classifying spaces of quandles, from results of group cohomology.

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Geometric Interpretations of Quandle Homology

Cited by 107 publications

References 15 publications

Quandle homotopy invariants of knotted surfaces

Quandle homotopy invariants of knotted surfaces

Twisted quandle homology theory and cocycle knot invariants

Homotopical interpretation of link invariants from finite quandles

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