On homotopy groups of quandle spaces and the quandle homotopy invariant of links

Nosaka, Takefumi

doi:10.1016/j.topol.2011.02.006

Cited by 31 publications

(54 citation statements)

References 22 publications

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“…For an X-coloring C ∈ Col X (D), we consider a sum τ ǫ τ (C(γ i ), C(γ j )) ∈ Z X × X , where τ runs over all crossing of D as shown in Figure 1 and the symbol ǫ τ ∈ {±1} denotes the sign of the crossing τ . As is known (see [RS,N1]), this sum is a 2-cycle, and homology classes of these sums in H Q 2 (X) are independent of Reidemeister moves and the concordance relations; Hence we obtain a homomorphism…”

Section: Review; Quandle Homotopy Invariant Of Linksmentioning

confidence: 77%

“…From the definitions of the maps H X and the 2-skeleton of the rack space BX, we can easily verify that this composite coincides with the Hurewicz map of BX modulo the direct summand Z O(X) (see [RS] and [N1,Proposition 3.12] for details). In conclusion, the formula (17) enables us to compute the Hurewicz map of BX.…”

Section: The Homomorphism θ Xmentioning

confidence: 98%

“…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: 99%

“…As an approach to the homotopy group π 2 (BX), the reader should keep in mind the following isomorphism shown by [FRS2] (see also [N1,Theorem 6.2] for the detailed description):…”

Section: Outline Of Proofs Of Theorem 35mentioning

confidence: 99%

“…where H is the Hurewicz map of BX and the τ is the transgression (see, e.g., [McC,§8.3 (14)), the restriction of the Hurewicz map H on the summand Z O(X) ⊂ π 2 (BX) is shown to be an isomorphism [N1,Proposition 3.12]. Therefore, recalling the isomorphism As(X) ∼ = π 1 (BX), the sequence (19) is reformulated as…”

Section: Outline Of Proofs Of Theorem 35mentioning

confidence: 99%

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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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Section: Review; Quandle Homotopy Invariant Of Linksmentioning

confidence: 77%

Section: The Homomorphism θ Xmentioning

confidence: 98%

Section: Review; Quandle Homotopy Invariant Of Linksmentioning

confidence: 99%

“…As an approach to the homotopy group π 2 (BX), the reader should keep in mind the following isomorphism shown by [FRS2] (see also [N1,Theorem 6.2] for the detailed description):…”

Section: Outline Of Proofs Of Theorem 35mentioning

confidence: 99%

Section: Outline Of Proofs Of Theorem 35mentioning

confidence: 99%

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Homotopical interpretation of link invariants from finite quandles

Nosaka

2015

Topology and its Applications

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Knot Quandle Decompositions

Cattabriga

Horvat

2020

Mediterr. J. Math.

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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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On homotopy groups of quandle spaces and the quandle homotopy invariant of links

Cited by 31 publications

References 22 publications

Homotopical interpretation of link invariants from finite quandles

Homotopical interpretation of link invariants from finite quandles

Knot Quandle Decompositions

Quandle homotopy invariants of knotted surfaces

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