Related papers: Homology for Quandles with Partial Group Operation…
The paper develops further the theory of quandle rings which was introduced by the authors in a recent work. Orderability of quandles is defined and many interesting examples of orderable quandles are given. It is proved that quandle rings…
Knot contact homology is an ambient isotopy invariant of knots and links in $\mathbb R^3$. The purpose of this paper is to extend this definition to an ambient isotopy invariant of tangles and prove that gluing of tangles gives a gluing…
Given a manifold with corners $X$, we associates to it the corner structure simplicial complex $\Sigma_X$. Its reduced K-homology is isomorphic to the K-theory of the $C^*$-algebra $\mathcal{K}_b(X)$ of b-compact operators on $X$. Moreover,…
This work arose from efforts to generalise the usual cubical boundary by using different 'weights' for opposite faces, but still to obtain a chain complex, and this method was found to generalise. We describe a variant of the classical…
Symmetric homology is an analog of cyclic homology in which the cyclic groups are replaced by symmetric groups. The foundations for the theory of symmetric homology of algebras are developed in the context of crossed simplicial groups using…
We construct a CW decomposition $C_n$ of the $n$-dimensional half cube in a manner compatible with its structure as a polytope. For each $3 \leq k \leq n$, the complex $C_n$ has a subcomplex $C_{n, k}$, which coincides with the clique…
Quandle colorings and cocycle invariants are studied for composite knots, and applied to chirality and abelian extensions. The square and granny knots, for example, can be distinguished by quandle colorings, so that a trefoil and its mirror…
In this paper we describe methods for computing rack and quandle cohomology. We illustrate these methods by completely determining the cohomology of prime dihedral quandles.
We give a first example of 2-knots with the same knot group but different knot quandles by analyzing the knot quandles of twist spins. As a byproduct of the analysis, we also give a classification of all twist spins with finite knot…
While knotoids on the sphere are well-understood by a variety of invariants, knotoids on the plane have proven more subtle to classify due to their multitude over knotoids on the sphere and a lack of invariants that detect a diagram's…
Coherence phenomena appear in two different situations. In the context of category theory the term `coherence constraints' refers to a set of diagrams whose commutativity implies the commutativity of a larger class of diagrams. In the…
Posed by Taniguchi, the classification of quandles with good involutions is a difficult question with applications to surface-knot theory. We address this question for subquandles of conjugation quandles, including all core quandles. We…
Topological gauge theories in four dimensions which admit surface operators provide a natural framework for realizing homological knot invariants. Every such theory leads to an action of the braid group on branes on the corresponding moduli…
We give a construction of quandle cocycles from group cocycles, especially, for any integer p \geq 3, quandle cocycles of the dihedral quandle R_p from group cocycles of the cyclic group Z/p. We will show that a group 3-cocycle of Z/p gives…
We define enhanced presentations of quandles via generators and relations with additional information comprising signed operations and an order structure on the set of generators. Such a presentation determines a virtual link diagram up to…
In this note, we survey our recent work concerning cohomologies of harmonic bundles on quasi-compact Kaehler manifolds.
For almost finite groupoids, we study how their homology groups reflect dynamical properties of their topological full groups. It is shown that two clopen subsets of the unit space has the same class in H_0 if and only if there exists an…
In a smooth dynamical system, a homoclinic connection is a closed orbit returning to a saddle equilibrium. Under perturbation, homoclinics are associated with bifurcations of periodic orbits, and with chaos in higher dimensions. Homoclinic…
We prove that if Q is a finite quasigroup quandle, then |Q| annihilates the torsion of its homology. It is a classical result in reduced homology of finite groups that the order of a group annihilates its homology. From the very beginning…
Homology with values in a connection with possibly irregular singular points on an algebraic curve is defined, generalizing homology with values in the underlying local system for a connection with regular singular points. Integration…