Trunks and classifying spaces

Fenn, Roger; Rourke, Colin; Sanderson, Brian

doi:10.1007/bf00872903

Cited by 179 publications

(200 citation statements)

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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%

“…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.…”

Section: Introductionmentioning

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: Reviews Of Rack Homology Quandle Homology and Topological Mmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Quandle homotopy invariants of knotted surfaces

Nosaka

2012

Math. Z.

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“…We derive a cohomology theory for quandles diagrammatically from Reidemeister moves for classical knots and knotted surfaces. Our definition of quandle (co)homology is a modification of rack (co)homology defined in [13] and [14]. Quandle cocycles are used to define state-sum invariants for knots and links in dimension 3 and for knotted surfaces in dimension 4.…”

Section: Introductionmentioning

confidence: 99%

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

Carter

Jelsovsky

Kamada

et al. 2003

Trans. Amer. Math. Soc.

336

456

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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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“…Next, we review the associated group [FRS1], denoted by As(X). This group is is defined by the group presentation…”

Section: Review Of Quandlesmentioning

confidence: 99%

“…Next, we briefly recall the quandle homotopy invariant of links (our formula is a modification the formula in [FRS1]). Let us consider the set, Π 2 (X), of all X-colorings of all diagrams subject to Reidemeister moves and the concordance relations illustrated in Figure 2.…”

Section: Review; Quandle Homotopy Invariant Of Linksmentioning

confidence: 99%

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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Trunks and classifying spaces

Cited by 179 publications

References 11 publications

Quandle homotopy invariants of knotted surfaces

Quandle homotopy invariants of knotted surfaces

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

Homotopical interpretation of link invariants from finite quandles

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