Enumerating finite racks, quandles and kei

A rack of order $n$ is a binary operation \rack on a set $X$ of cardinality $n$, such that right multiplication is an automorphism. More precisely, (X,\rack) is a rack provided that the map x\mapsto x\rack y is a bijection for all $y\in X$, and (x\rack y)\rack z=(x\rack z)\rack (y\rack z) for all $x,y,z\in X$. The paper provides upper and lower bounds of the form $2^{cn^2}$ on the number of isomorphism classes of racks of order $n$. Similar results on the number of isomorphism classes of quandles and kei are obtained. The results of the paper are established by first showing how an arbitrary rack is related to its operator group (the permutation group on $X$ generated by the maps x\mapsto x\rack y for $y\in Y$), and then applying some of the theory of permutation groups. The relationship between a rack and its operator group extends results of Joyce and of Ryder; this relationship might be of independent interest.Comment: 11 page

Rack shadows and their invariants

A rack shadow is a set X with a rack action by a rack R, analogous to a vector space over a field. We use shadow colorings of classical link diagrams to define enhanced rack counting invariants and show that the enhanced invariants are stronger than unenhanced counting invariants.Comment: 9 page

A universal enveloping algebra for cocommutative rack bialgebras

We construct a bialgebra object in the category of linear maps LM from a cocommutative rack bialgebra. The construction does extend to some non-cocommutative rack bialgebras, as is illustrated by a concrete example. As a separate result, we show that the Loday complex with adjoint coefficients embeds into the rack bialgebra deformation complex for the rack bialgebra defined by a Leibniz algebra.Comment: 17 page

The rack space

The main result of this paper is a new classification theorem for links (smooth embeddings in codimension 2). The classifying space is the rack space (defined in [Trunks and classifying spaces, Applied Categorical Structures, 3 (1995) 321--356]) and the classifying bundle is the first James bundle (defined in "James bundles" math.AT/0301354). We investigate the algebraic topology of this classifying space and report on calculations given elsewhere. Apart from defining many new knot and link invariants (including generalised James--Hopf invariants), the classification theorem has some unexpected applications. We give a combinatorial interpretation for \pi_2 of a complex which can be used for calculations and some new interpretations of the higher homotopy groups of the 3--sphere. We also give a cobordism classification of virtual links.Comment: This paper is largely extracted from our January 1996 preprint `James bundles and applications' available at http://www.maths.warwick.ac.uk/~cpr/ftp/james.ps Version 2: minor correction

Rack invariants of links in L(p,1)L(p,1)

We describe a presentation for the augmented fundamental rack of a link in the lens space $L(p,1)$. Using this presentation, the (enhanced) counting rack invariants that have been defined for the classical links are applied to the links in $L(p,1)$. In this case, the counting rack invariants also include the information about the action of $\pi_{1}(L(p,1))$ on the augmented fundamental rack of a link

Rack and quandle homology

The theory of rack and quandle modules is developed - in particular a tensor product is defined, and shown to satisfy an appropriate adjointness condition. Notions of free rack and quandle modules are introduced, and used to define an enveloping object (the `rack algebra' or `wring') for a given rack or quandle. These constructions are then used to define homology and cohomology theories for racks and quandles which contain all currently-known variants.Comment: 16 pages, LaTeX2e. Requires gtart.cls and diagram.sty (included

Locality-Aware Hybrid Coded MapReduce for Server-Rack Architecture

MapReduce is a widely used framework for distributed computing. Data shuffling between the Map phase and Reduce phase of a job involves a large amount of data transfer across servers, which in turn accounts for increase in job completion time. Recently, Coded MapReduce has been proposed to offer savings with respect to the communication cost incurred in data shuffling. This is achieved by creating coded multicast opportunities for shuffling through repeating Map tasks at multiple servers. We consider a server-rack architecture for MapReduce and in this architecture, propose to divide the total communication cost into two: intra-rack communication cost and cross-rack communication cost. Having noted that cross-rack data transfer operates at lower speed as compared to intra-rack data transfer, we present a scheme termed as Hybrid Coded MapReduce which results in lower cross-rack communication than Coded MapReduce at the cost of increase in intra-rack communication. In addition, we pose the problem of assigning Map tasks to servers to maximize data locality in the framework of Hybrid Coded MapReduce as a constrained integer optimization problem. We show through simulations that data locality can be improved considerably by using the solution of optimization to assign Map tasks to servers.Comment: 5 pages, accepted to IEEE Information Theory Workshop (ITW) 201

Complemented lattices of subracks

In this paper, a question due to Heckenberger, Shareshian and Welker on racks in [7] is positively answered. A rack is a set together with a selfdistributive bijective binary operation. We show that the lattice of subracks of every finite rack is complemented. Moreover, we characterize finite modular lattices of subracks in terms of complements of subracks. Also, we introduce a certain class of racks including all finite groups with the conjugation operation, called G- racks, and we study some of their properties. In particular, we show that a finite G-rack has the homotopy type of a sphere. Further, we show that the lattice of subracks of an infinite rack is not necessarily complemented which gives an affirmative answer to the aformentioned question. Indeed, we show that the lattice of subracks of the set of rational numbers, as a dihedral rack, is not complemented. Finally, we show that being a Boolean algebra, pseudocomplemented and uniquely complemented as well as distributivity are equivalent for the lattice of subracks of a rack

Structure theory of Rack-Bialgebras

In this paper we focus on a certain self-distributive multiplication on coalgebras, which leads to so-called rack bialgebra. Inspired by semi-group theory (adapting the Suschkewitsch theorem), we do some structure theory for rack bialgebras and cocommutative Hopf dialgebras. We also construct canonical rack bialgebras (some kind of enveloping algebras) for any Leibniz algebra and compare to the existing constructions. We are motivated by a differential geometric procedure which we call the Serre functor: To a pointed differentible manifold with multiplication is associated its distribution space supported in the chosen point. For Lie groups, it is well-known that this leads to the universal enveloping algebra of the Lie algebra. For Lie racks, we get rack-bialgebras, for Lie digroups, we obtain cocommutative Hopf dialgebras.Comment: 59 pages. The initial article was split during refereeing. This is the part on the structure theory of rack bialgebra

On rack polynomials

We study rack polynomials and the link invariants they define. We show that constant action racks are classified by their generalized rack polynomials and show that $ns^at^a$-quandles are not classified by their generalized quandle polynomials. We use subrack polynomials to define enhanced rack counting invariants, generalizing the quandle polynomial invariants.Comment: 9 page