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Idempotents, free products and quandle coverings
Abstract
In this paper, we investigate idempotents in quandle rings and relate them with quandle coverings. We prove that integral quandle rings of quandles of finite type that are nontrivial coverings over nice base quandles admit infinitely many nontrivial idempotents, and give their complete description. We show that the set of all these idempotents forms a quandle in itself. As an application, we deduce that the quandle ring of the knot quandle of a nontrivial long knot admit nontrivial idempotents. We consider free products of quandles and prove that integral quandle rings of free quandles have only trivial idempotents, giving an infinite family of quandles with this property. We also give a description of idempotents in quandle rings of unions and certain twisted unions of quandles.
... Since their introduction, quandles and their kin (biquandles, racks and biracks) have been used extensively to obtain invariants of knots in the 3-space and knotted surfaces in 4-space (see for example [4][5][6]). There have been also interest in quandles and their relations to other areas of mathematics such as ring theory [7][8][9][10], quasigroups and Moufang loops [11], representation theory [12,13], Lie algebras [14,15], Yang-Baxter equation [16], Hopf algebras [15,17] and Frobenius algebras [18]. ...
This paper gives a new way of characterizing L-space 3-manifolds by using orderability of quandles. Hence, this answers a question of Clay et al. (Question 1.1 of Can Math Bull 59(3):472–482, 2016). We also investigate both the orderability and circular orderability of dynamical extensions of orderable quandles. We give conditions under which the conjugation quandle on a group, as an extension of the conjugation of a bi-orderable group by the conjugation of a right orderable group, is right orderable. We also study the right circular orderability of link quandles. We prove that the n-quandle Qn(L) of the link quandle of a link L in the 3-sphere is not right circularly orderable and hence it is not right orderable. But on the other hand, we show that there are infinitely many links for which the p-enveloping group of the link quandle is right circularly orderable for any prime integer p.
... Since their introduction, quandles and their kin (biquandles, racks and biracks) have been used extensively to obtain invariants of knots in the 3-space and knotted surfaces in 4-space (see for example [10,15,23]). There have been also interest in quandles and their relations to other areas of mathematics such as ring theory [5,20,24,25], quasigroups and Moufang loops [19], representation theory [21,26], Lie algebras [11,12], Yang-Baxter equation [14], Hopf algebras [1,12] and Frobenius algebras [13]. ...
This paper gives a new way of characterizing L-space 3-manifolds by using orderability of quandles. Hence, this answers a question of Adam Clay et al. [Question 1.1 of Canad. Math. Bull. 59 (2016), no. 3, 472-482]. We also investigate both the orderability and circular orderability of dynamical extensions of orderable quandles. We give conditions under which the conjugation quandle on a group, as an extension of the conjugation of a bi-orderable group by the conjugation of a right orderable group, is right orderable. We also study the right circular orderability of link quandles. We prove that the n-quandle of the link quandle of L is not right circularly orderable and hence it is not right orderable. But on the other hand, we show that there are infinitely many links for which the p-enveloping group of the link quandle is right circularly orderable for any prime integer p.
This article serves two purposes. The first is to give an introduction to the readers who are not so familiar with quandle theory. The second is to report on new results. The first new result is that we compute second and third homology groups of disjoint union of quandles using the machinery of spectral sequences. We also establish a connection between graph theory and quandle theory via quandle rings. More specifically, we prove that for a finite commutative Latin quandle, the zero-divisor graph of the quandle ring with binary coefficients is connected and has a diameter bounded by two below and bounded by four above. We also investigate graph automorphisms of quandle rings and their relations with automorphisms of quandle rings and automorphisms of quandles. In particular we compute the ring automorphism groups of the quandle rings of the dihedral quandle of four elements and the trivial quandles of orders two, three and four.
In this paper, we develop a theory of rack (respectively, quandle) rings analogous to the classical theory of group rings for groups. Let X be a rack and R an associative ring with unity. We introduce the rack ring R[X] of X with coefficients in R, and show that the ring R[X] gives interesting information about the rack X. We define the augmentation ideal of R[X], and show that the quandle X is trivial if and only if . Further, we develop a dictionary between subquandles of the quandle X and ideals of the associated quandle ring R[X]. Since the ring R[X] does not have unity, we introduce the extended rack ring , which is a ring with unity, and obtain some description of unit groups in . Finally, we obtain some results on the power-associativity of quandle rings of dihedral quandles.
In this paper we study different questions concerning automorphisms of quandles. For a conjugation quandle of a group G we determine several subgroups of and find necessary and sufficient conditions when these subgroups coincide with the whole group . In particular, we prove that if and only if either or G is one of the groups , or . For a big list of Takasaki quandles T(G) of an abelian group G with 2-torsion we prove that the group of inner automorphisms is a Coxeter group. We study automorphisms of certain extensions of quandles and determine some interesting subgroups of the automorphism groups of these quandles. Also we classify finite quandles Q with -transitive action of .
A homology theory is developed for set-theoretic Yang-Baxter equations, and knot invariants are constructed by generalized colorings by biquandles and Yang-Baxter cocycles.
The purpose of this paper is to introduce and investigate the notion of derivation for quandle algebras. More precisely, we describe the symmetries on structure constants providing a characterization for a linear map to be a derivation. We obtain a complete characterization of derivations in the case of quandle algebras of dihedral quandles over fields of characteristic zero, and provide the dimensionality of the Lie algebra of derivations. Many explicit examples and computations are given over both zero and positive characteristic. Furthermore, we investigate inner derivations, in the sense of Schafer for non-associative structures. We obtain necessary conditions for the Lie transformation algebra of quandle algebras of Alexander quandles, with explicit computations in low dimensions.
We associate to every quandle X and an associative ring with unity k, a nonas-sociative ring k[X] following Bardakov et al. (Quandle rings, arXiv:1709.03069, [3]). The basic properties of such rings are investigated. In particular, under the assumption that the inner automorphism group Inn(X) acts orbit 2-transitively on X, a complete description of right (or left) ideals is provided. The complete description of right ideals for the dihedral quandles R n is given. It is also shown that if for two quandles X and Y the inner automor-phism groups act 2-transitively and k[X] is isomorphic to k[Y], then the quandles are of the same partition type. However, we provide examples when the quandle rings k[X] and k[Y] are isomorphic, but the quandles X and Y are not isomorphic. These examples answer some open problems in the article of Bardakov et al.([3]).
In this paper we introduce distinct approaches to loop braid groups, a generalisation of braid groups, and unify all the definitions that have appeared so far in literature, with a complete proof of the equivalence of these definitions. These groups have in fact been an object of interest in different domains of mathematics and mathematical physics, and have been called, in addition to loop braid groups, with several names such as of motion groups, groups of permutation-conjugacy automorphisms, braid-permutation groups, welded braid groups and untwisted ring groups. In parallel to this, we introduce an extension of these groups that appears to be a more natural generalisation of braid groups from the topological point of view. Throughout the text we motivate the interest in studying loop braid groups and give references to some of their applications.
We consider a groupoid structure involving graphs and imbedded circles (resp. disks) in a closed orientable surface F (resp. the handlebody H it bounds). We look at generalized connected sums and related operations "induced" by certain morphisms, and introduce a quandle structure involving isotopy classes of imbedded circles and Dehn twists in F, Some relations between these operations are examined, and we look at characterizations of imbedded disks in H arising in this algebraic context.
Isotopy classes of circles on an orientable surface F of genus g form a quandle Q under the operation of Dehn twisting about such circles. We derive certain fundamental relations in the Dehn quandle and then consider a homology theory based on this quandle. We show how certain types of relations in the quandle translate into cycles and homology representatives in this homology theory, and characterize a large family of 2 -cycles representing homology elements. Finally we draw connections to Lefschetz fibrations, showing isomorphism classes of such fibrations over a disk correspond to quandle homology classes in dimension 2, and discuss some further structures on the homology.
A fundamental step in the classification of finite-dimensional complex pointed Hopf algebras is the determination of all finite-dimensional Nichols algebras of braided vector spaces arising from groups. The most important class of braided vector spaces arising from groups is the class of braided vector spaces , where X is a rack and q is a 2-cocycle on X with values in . Racks and cohomology of racks appeared also in the work of topologists. This leads us to the study of the structure of racks, their cohomology groups and the corresponding Nichols algebras. We will show advances in these three directions. We classify simple racks in group-theoretical terms; we describe projections of racks in terms of general cocycles; we introduce a general cohomology theory of racks containing properly the existing ones. We introduce a “Fourier transform” on racks of certain type; finally, we compute some new examples of finite-dimensional Nichols algebras.
Given a knot K in the 3-sphere, let QK be its fundamental quandle as introduced by Joyce. Its first homology group is easily seen to be . We prove that H2(QK)=0 if and only if K is trivial, and whenever K is non-trivial. An analogous result holds for links, thus characterizing trivial components.More detailed information can be derived from the conjugation quandle: let QKπ be the conjugacy class of a meridian in the knot group . We show that , where p is the number of prime summands in a connected sum decomposition of K.
We consider the subgroup of the automorphism group of the free group generated by the braid group and the permutation group. This is proved to be the same as the subgroup of automorphisms of permutation-conjugacy type and is represented by generalised braids (braids in which some crossings are allowed to be “welded”). As a consequence of this representation there is a finite presentation which shows the close connection with both the classical braid and permutation groups. The group is isomorphic to the automorphism group of the free quandle and closely related to the automorphism group of the free rack. These automorphism groups are connected with invariants of classical knots and links in the 3-sphere.
This article establishes the algebraic covering theory of quandles. For every connected quandle we explicitly construct a universal covering, which in turn leads us to define the algebraic fundamental group as the automorphism group of the universal covering. We then establish the Galois correspondence between connected coverings and subgroups of the fundamental group. Quandle coverings are thus formally analogous to coverings of topological spaces, and resemble Kervaire's algebraic covering theory of perfect groups. A detailed investigation also reveals some crucial differences, which we illustrate by numerous examples. As an application we obtain a simple formula for the second (co)homology group of a quandle Q. It has long been known that H_1(Q) = H^1(Q) = \Z[\pi_0(Q)], and we construct natural isomorphisms H_2(Q) = \pi_1(Q,q)_{ab} and H^2(Q,A) = Ext(Q,A) = Hom(\pi_1(Q,q),A), reminiscent of the classical Hurewicz isomorphisms in degree 1. This means that whenever the fundamental group is known, (co)homology calculations in degree 2 become very easy.
Given a rack Q and a ring A, one can construct a Yang-Baxter operator c_Q: V tensor V --> V tensor V on the free A-module V = AQ by setting c_Q(x tensor y) = y tensor x^y for all x,y in Q. In answer to a question initiated by D.N. Yetter and P.J. Freyd, this article classifies formal deformations of c_Q in the space of Yang-Baxter operators. For the trivial rack, where x^y = x for all x,y, one has, of course, the classical setting of r-matrices and quantum groups. In the general case we introduce and calculate the cohomology theory that classifies infinitesimal deformations of c_Q. In many cases this allows us to conclude that c_Q is rigid. In the remaining cases, where infinitesimal deformations are possible, we show that higher-order obstructions are the same as in the quantum case.