Related papers: Rack and quandle homology
In this paper, we explore the bounded cohomology of quandles and its applications to knot theory. We establish two key results that provide sufficient conditions for the infinite dimensionality of the second bounded cohomology of quandles.…
Quandle homology was defined from rack homology as the quotient by a subcomplex corresponding to the idempotency, for invariance under the type I Reidemeister move. Similar subcomplexes have been considered for various identities of racks…
We restructure and advance the classification theory of finite racks and quandles by employing powerful methods from transformation groups and representation theory, especially Burnside rings. These rings serve as universal receptacles for…
A quandle is a set that has a binary operation satisfying three conditions corresponding to the Reidemeister moves. Homology theories of quandles have been developed in a way similar to group homology, and have been applied to knots and…
In this article, we characterize the (covariant) isotropy groups of free, finitely generated racks and quandles. As a consequence, we show that the usual inner automorphisms of such racks and quandles are precisely those automorphisms that…
A topological commutative ring is said to be rigid when for every set $X$, the topological dual of the $X$-fold topological product of the ring is isomorphic to the free module over $X$. Examples are fields with a ring topology, discrete…
In this paper, motivated by a work of Luk and Yau, and Huneke and Wiegand, we study various aspects of the cohomological rigidity property of tensor product of modules over commutative Noetherian rings. We determine conditions under which…
We found a necessary and sufficient condition for the existence of the tensor product of modules over a vertex algebra. We defined the notion of vertex bilinear map and we provide two algebraic construction of the tensor product, where one…
A continuous cohomology theory for topological quandles is introduced, and compared to the algebraic theories. Extensions of topological quandles are studied with respect to continuous 2-cocycles, and used to show the differences in second…
We define an invariant of tangles and framed tangles given a finite crossed module and a pair of functions, called a Reidemeister pair, satisfying natural properties. We give several examples of Reidemeister pairs derived from racks,…
The quandle homology theory is generalized to the case when the coefficient groups admit the structure of Alexander quandles, by including an action of the infinite cyclic group in the boundary operator. Theories of Alexander extensions of…
Lower bounds of betti numbers for homology groups of racks and quandles will be given using the quotient homomorphism to the orbit quandles. Exact sequences relating various types of homology groups are analyzed. Geometric methods of…
The paper establishes new relationship between cohomology, extensions and automorphisms of quandles. We derive a four term exact sequence relating quandle 1-cocycles, second quandle cohomology and certain group of automorphisms of an…
We associate to every quandle $X$ and an associative ring with unity $\mathbf{k}$, a nonassociative ring $\mathbf{k}[X]$ following [3]. The basic properties of such rings are investigated. In particular, under the assumption that the inner…
A quandle is an algebra with two binary operations satisfying three conditions which are related to Reidemeister moves in knot theory. In this paper we introduce the notion of the (canonical) tensor product of a quandle. The tensor product…
We study rack and quandle coverings from a universal algebraic viewpoint and we show how they can be understood using the notion of strongly abelian congruences. We provide an abstract characterization of several particular types of…
Algebraic homology and cohomology theories for quandles have been studied extensively in recent years. With a given quandle 2(3)-cocycle one can define a state-sum invariant for knotted curves(surfaces). In this paper we introduce another…
Quandles can be regarded as generalizations of symmetric spaces. In the study of symmetric spaces, the notion of flatness plays an important role. In this paper, we define the notion of flat quandles, by referring to the theory of…
We define an invariant of tangles and framed tangles given a finite crossed module and a pair of functions, called a Reidemeister pair, satisfying natural properties. We give several examples of Reidemeister pairs derived from racks,…
In this paper, we aim to provide a notion of "relative objects", i.e. objects equipped with some sort of subobjects, in differential topology. In spite of active researches relating them, e.g. knot theory or the theory of manifolds with…