On quandle homology groups of Alexander quandles of prime order

Nosaka, Takefumi

doi:10.1090/s0002-9947-2013-05754-6

Cited by 30 publications

(19 citation statements)

References 13 publications

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“…is shown to be annihilated by q (see [24,Corollary 6.2]), we now show H 2 Q (X ; F q ) ∼ = 0 for the proof of H Q 2 (X ; Z) ∼ = 0. To this end, it suffices to show ω p i +1 = 1 for any i ≤ 2h by (21).…”

Section: Proofs Of Theorem 39 and Proposition 34mentioning

confidence: 51%

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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: Proofs Of Theorem 39 and Proposition 34mentioning

confidence: 51%

“…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].…”

Section: Introductionmentioning

confidence: 99%

Quandle homotopy invariants of knotted surfaces

Nosaka

2012

Math. Z.

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“…The above result was proposed as a conjecture in [NP2] and special cases of the above theorem was proven in [Cla,Nos1].…”

Section: Rack and Quandle Homologymentioning

confidence: 78%

On the rack homology of graphic quandles

Mukherjee¹,

Przytycki²

2019

Nonassociative Mathematics and Its Applications

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This paper has partially a novel and partially a survey character. We start with a short review of rack (two term) homology of self distributive algebraic structures (shelves) and their connections to knot theory. We concentrate on a sub-family of quandles satisfying the graphic axiom. For a large family of graphic quandles (including infinite ones), we compute the second rack homology groups. Finally, we propose conjectures based on our computational data.Quandles are algebraic structures with axioms motivated by the Reidemeister moves from knot theory ([Joy, Mat]). More general algebraic structures such as racks, spindles, and shelves are obtained by discarding some of the axioms of a quandle. A number of homology theories related to quandles have been developed in the last three decades with connections to knot theory. Rack (two term) homology was introduced in [FRS1, FRS2, FRS3] and was later enhanced into quandle homology in [CJKLS] to define quandle cocycle invariants for classical knots and links. While for rack homology it is enough to work with shelves, for quandle homology, spindles are necessary (see Section 1).

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“…In [23] it is conjectured that for a finite quasigroup quandle, torsion of its homology is annihilated by the order of the quandle. The conjecture is proved by T. Nosaka for finite Alexander quasigroup quandles [24] (see also [8] for the case of dihedral quandles of prime order).…”

Section: History Of the Problemmentioning

confidence: 97%