Quandle homotopy invariants of knotted surfaces

Abstract

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...

The geometric realization of a normalized set-theoretic Yang-Baxter homology of biquandles

“…The rack homotopy invariant of framed oriented links, obtained from the classifying spaces of racks, was introduced previously [9]. It was transformed into the quandle homotopy invariant [20,21] and the shadow homotopy invariant [29] of oriented links using the classifying spaces of quandles, which are constructed by adding extra cells to the classifying spaces of racks. In a similar manner, we construct a homotopy invariant of oriented links using the classifying spaces of biquandles.…”

“…(cf. [20,29,21]). By using Proposition 3.1 and Proposition 3.2, one can construct homological and homotopical link invariants in a similar manner to [4,20,21].…”

“…[20,29,21]). By using Proposition 3.1 and Proposition 3.2, one can construct homological and homotopical link invariants in a similar manner to [4,20,21]. For example, we define the homotopical state-sum invariants of oriented links and oriented closed knotted surfaces as follows:…”

“…Meanwhile, Joyce [12] and Matveev [17] independently introduced a self-distributive algebraic structure, called a quandle, which satisfies axioms motivated by the Reidemeister moves, and it has been generalized as a biquandle. Quandles and biquandles are solutions of the set-theoretic Yang-Baxter equation, which have been used to define homotopical and homological invariants of knots and links [20,21,29,2,4,3]. This paper describes the study of a normalized homology theory of a set-theoretic solution of the Yang-Baxter equation and its geometric realization.…”

Cyclic branched coverings of knots and quandle homology

“…We remark that when p is prime, dim ‫ކ‬ p H n Q (R p ; ‫ކ‬ p ) was calculated for any n by Nosaka [2009], who gave a system of generators of H n Q (R p ; ‫ކ‬ p ). When p is an odd integer, Nosaka [2010] showed that…”