On pallets for Fox colorings of spatial graphs

We introduce the notion of pallets of quandles and define coloring invariants for spatial graphs which give a generalization of Fox colorings studied in Ishii and Yasuhara (1997) [4]. All pallets for dihedral quandles are obtained from the quotient sets of the universal pallets under a certain equivalence relation. We study the quotient sets and classify their elements.     

Higher Coverings of Racks and Quandles—Part II

Applied Categorical Structures - This article is the second part of a series of three articles, in which we develop a higher covering theory of racks and quandles. This project is rooted in...     

General constructions of biquandles and their symmetries

Biquandles are algebraic objects with two binary operations whose axioms encode the generalized Reidemeister moves for virtual knots and links. These objects also provide set theoretic solutions of the well-known Yang-Baxter equation. The first half of this paper proposes some natural constructions of biquandles from groups and from their simpler counterparts, namely, quandles. We completely determine all words in the free group on two generators that give rise to (bi)quandle structures on all groups. We give some novel constructions of biquandles on unions and products of quandles, including what we refer as the holomorph biquandle of a quandle. These constructions give a wealth of solutions of the Yang-Baxter equation. We also show that for nice quandle coverings a biquandle structure on the base can be lifted to a biquandle structure on the covering. In the second half of the paper, we determine automorphism groups of these biquandles in terms of associated quandles showing elegant relationships between the symmetries of the underlying structures.     

Canonical Forms for Operation Tables of Finite Connected Quandles

We introduce a notion of natural orderings of elements of connected finite quandles. Let Q be such a quandle of order n. Any automophism on Q is a natural ordering when the elements are already ordered naturally. Suppose that the permutation *q is a cycle of length n ? 1. Then, the operation tables for such orderings coincide, which leads to the classification of automorphisms on Q. Moreover, every row and column of the operation table contains all the elements of such a quandle, which is due to K. Oshiro. We also consider the general case of finite connected quandles.     

Complexity of a knot invariant

The algebraic structures called quandles constitute a complete invariant for tame knots. However, determining when two quandles are isomorphic is an empirically hard problem, so there is some dissatisfaction with quandles as knot invariants. We have confirmed this apparent difficulty, showing within the framework of Borel reducibility that the general isomorphism problem for quandles is as complex as possible. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Principal and doubly homogeneous quandles

Monatshefte für Mathematik - In the paper we describe the class of principal quandles and we show that connected quandles can be decomposed as a disjoint union of principal quandles. We also...     

Subdirectly irreducible medial quandles

We describe all subdirectly irreducible medial quandles. We show that they fall within one of four disjoint classes. In particular, in the finite case they are either connected (and therefore Alexander quandles) or reductive. Moreover, we provide a representation of all non-connected subdirectly irreducible medial quandles.     

Free medial quandles

This paper gives the construction of free medial quandles as well as free n-symmetric medial quandles and free m-reductive medial quandles.     

Some calculations of cohomology groups of finite Alexander quandles

We calculate some quandle cohomology groups; the rational cohomology groups of any finite Alexander quandles, the second cohomology groups with a finite field coefficient of any finite Alexander quandles over a finite fields, and the third cohomology groups of the finite Alexander quandles of the form .     

Permutations,power operations,and the center of the category of racks

Racks and quandles are rich algebraic structures that are strong enough to classify knots. Here we develop several fundamental categorical aspects of the theories of racks and quandles and their relation to the theory of permutations. In particular, we compute the centers of the categories and describe power operations on them, thereby revealing free extra structure that is not apparent from the definitions. This also leads to precise characterizations of these theories in the form of universal properties.     

Quandle cohomology and state-sum invariants of knotted curves and surfaces

The 2-twist spun trefoil is an example of a sphere that is knotted in 4-dimensional space. A proof is given in this paper that this sphere is distinct from the same sphere with its orientation reversed. Our proof is based on a state-sum invariant for knotted surfaces developed via a cohomology theory of racks and quandles (also known as distributive groupoids).

A quandle is a set with a binary operation -- the axioms of which model the Reidemeister moves in classical knot theory. Colorings of diagrams of knotted curves and surfaces by quandle elements, together with cocycles of quandles, are used to define state-sum invariants for knotted circles in -space and knotted surfaces in -space.

Cohomology groups of various quandles are computed herein and applied to the study of the state-sum invariants. Non-triviality of the invariants is proved for a variety of knots and links, and conversely, knot invariants are used to prove non-triviality of cohomology for a variety of quandles.

     

Finitely stable racks and rack representations

We define a new class of racks, called finitely stable racks, which, to some extent, share various flavors with Abelian groups. Characterization of finitely stable Alexander quandles is established. Further, we study twisted rack dynamical systems, construct their cross-products, and introduce representation theory of racks and quandles. We prove several results on the strong representations of finite connected involutive racks analogous to the properties of finite Abelian groups. Finally, we define the Pontryagin dual of a rack as an Abelian group which, in the finite involutive connected case, coincides with the set of its strong irreducible representations.     

Construction of Right Universal Multiplication Groups of Right Quasigroups

The right universal multiplication group of a quandle in the variety of all quandles is constructable from the right Cayley graph of that quandle.AMS Subject Classification (2000), 20N     

The Betti numbers of some finite racks

We show that the lower bounds for Betti numbers given in (J. Pure Appl. Algebra 157 (2001) 135) are equalities for a class of racks that includes dihedral and Alexander racks. We confirm a conjecture from the same paper by defining a splitting for the short exact sequence of quandle chain complexes. We define isomorphisms between Alexander racks of certain forms, and we also list the second and third homology groups of some dihedral and Alexander quandles.     

The quandles with two operations

Virtual quandles with two operations and the corresponding invariants of long virtual knots are discussed. A certain knot invariant is constructed and an example of proof that two knots are not equivalent in terms of this invariant is presented.     

Endofunctors of quandles and racks

We show that the only endofunctors of the category of quandles commuting with the forgetful functor to sets are the power operations. We also give a similar statement for racks.     

On Finite Racks and Quandles

We revisit finite racks and quandles using a perspective based on permutations which can aid in the understanding of the structure. As a consequence we recover old results and prove new ones. We also present and analyze several examples.

Communicated by M. Dixon.     

Covering Sequences of Boolean Functions and Their Cryptographic Significance

We introduce the notion of covering sequence of a Boolean function, related to the derivatives of the function. We give complete characterizations of balancedness, correlation immunity and resiliency of Boolean functions by means of their covering sequences. By considering particular covering sequences, we define subclasses of (correlation-immune) resilient functions. We derive upper bounds on their algebraic degrees and on their nonlinearities. We give constructions of resilient functions belonging to these classes. We show that they achieve the best known trade-off between order of resiliency, nonlinearity and algebraic degree.     

Quandle homology and complex volume

We introduce a new homology theory of quandles, called simplicial quandle homology, which is quite different from quandle homology developed by Carter et al. We construct a homomorphism from a quandle homology group to a simplicial quandle homology group. As an application, we obtain a method for computing the complex volume of a hyperbolic link only from its diagram.     

On the Brouwer Dimension of Compact Spaces

A closed set is called a cut between two disjoint sets if any continuum intersecting both these sets intersects the cut. The main result of this paper is that, for any compact space, the dimension defined by induction on the basis of the notion of cut does not exceed the covering dimension.