The Betti numbers of some finite racks

2012

Math. Z.

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...

“…By (5), we have H R 2 (X ; Z) ∼ = Z. It is known [5,17] that, for any x ∈ X , the 2-cycle of the form…”

Section: Proofs Of Lemmas 58 and 57mentioning

confidence: 99%

“…Furthermore, the above sequence enables us to compute π 2 (B X). For several quandles X of order ≤ 9, we determine π 2 (B X) exactly (see Table 1), following the values of H Q 2 (X ; Z) and H Q 3 (X ; Z) presented in [17]. Furthermore, since Mochizuki [19,20] …”

mentioning

confidence: 99%

Quandle homotopy invariants of knotted surfaces

2012

Math. Z.

“…where the first isomorphism is derived from Remark 5.1, and the second was shown [LN,Theorem 2.2]. Composing this (28) with the result on π 2 (BX) = π 2 (B(X, X)) from Theorem 3.5 can compute some torsion of the quandle homology H Q 3 (X) [see Lemma (8.4)].…”

Section: Application To Third Quandle Homologiesmentioning

confidence: 93%

Homotopical interpretation of link invariants from finite quandles

Topology and its Applications

2015

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.

“…As a special case, if X is the Alexander quandle of order p, then H Q n (X; Z) is annihilated by p (Corollary 6.4), proving [7,Conjecture 16]. It is known [4,Theorem.1] that H Q n (X; Z) is annihilated by |X| n for a connected quandle X and each n ≥ 1. Then Corollary 6.2 is a stronger estimate for Alexander quandles, while it does not hold for a connected quandles; for example, there exists a connected non-Alexander quandle QS (6) whose third quandle homology is not annihilated by |QS(6)| (Remark 6.3).…”

Section: Introductionmentioning

confidence: 94%

“…T. Mochizuki listed all 2-cocycles for finite Alexander quandles over a finite field in [5] and all 3-cocycles for Alexander quandles on a finite field in [6]. R. A. Litherland and S. Nelson analyzed the free and torsion subgroup of the quandle homology group of a finite quandle [4]. For the quandle homology of higher degrees, M. Niebrzydowski and J. H. Przytycki constructed some quandle homological operations and estimated the torsion subgroup of the integral quandle homology groups of some quandles.…”

Section: Introductionmentioning

confidence: 99%

On quandle homology groups of Alexander quandles of prime order

Trans. Amer. Math. Soc.

2013

In this paper we determine the integral quandle homology groups of Alexander quandles of prime order. As a special case, this settles the delayed F ibonacci conjecture by M. Niebrzydowski and J. H. Przytycki in [7]. Moreover, we determine the cohomology group of the Alexander quandle and obtain relatively simple presentations of all higher degree cocycles which generate the cohomology group. Furthermore, we prove that the integral quandle homology of a finite connected Alexander quandle is annihilated by the order of the quandle.