arXiv research

T. Mochizuki determined all 3-cocycles of the third quandle cohomologies of Alexander quandles on finite fields. We show that all the 3-cocycles, except those of 2-cocycle forms, are derived from group 3-cocycles of a meta-abelian group. Further, the quandle cocycle invariant of a link using Mochizuki's 3-cocycle is eq…

Study lengths of 3-cocycles for specific quandles, finding knot properties.

New invariant for spin 3-manifolds using super 3-cocycles.

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 rise to a non-trivial quandle 3-cocycle of R_p. When p is an odd prime, since d…

New algebraic rules for 5D shapes based on 3D cocycles.

This paper calculates Dijkgraaf-Witten invariants from Chern classes.

This paper exhibits equivalences of 2-stacks between certain models of $\mathbb{S}^1$-gerbes and differential 3-cocycles. We focus primarily on the model of Dixmier-Douady bundles, and provide an equivalence between the 2-stack of Dixmier-Douady bundles and the 2-stack of differential 3-cocycles of height 1, where the …

We determine the third cohomology of Alexander quandles of the form F_q[T]/(T-omega), where F_q denotes the finite field of order q and omega is an element of F-q which is neither 0 nor 1. As a result, we obtain many concrete examples of non-trivial 3-cocycles.

We focus on the cohomology of the $k$-th nilpotent quotient of the free group, $F/F_k$. This paper describes all the group 2-, 3-cocycles in terms of Massey products, and gives expressions for some of the 3-cocycles. We also give simple proofs of some of the results on Milnor invariants and the Johnson-Morita homomorph…

New shifting chain map enhances quandle invariants for links.

Recent work applying higher gauge theory to the superstring has indicated the presence of `higher symmetry'. Infinitesimally, this is realized by a `Lie 2-superalgebra' extending the Poincare superalgebra in precisely the dimensions where the classical supersymmetric string makes sense: 3, 4, 6 and 10. In the previous …

In this paper, we introduce the (co)homology group of a multiple conjugation biquandle. It is the (co)homology group of the prismatic chain complex, which is related to the homology of foams introduced by J. S. Carter, modulo a certain subchain complex. We construct invariants for $S^1$-oriented handlebody-links using …

We define the sigma-model action for world-sheets with embedded defect networks in the presence of a three-form field strength. We derive the defect gluing condition for the sigma-model fields and their derivatives, and use it to distinguish between conformal and topological defects. As an example, we treat the WZW mod…

The purpose of this contribution is to point out connections between recent ideas about gerbes and gerbal actions (as higher categorical extension of representation theory) and old discussion in quantum field theory on commutator anomalies, gauge group extensions, and 3-cocycles. The unifying concept is the classical o…

Paper studies non-associativity in quantum systems with magnetic fields.

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 version of quandle (co)homology theory, say positive quandle (co)homology. Some p…

Unified invariant for immersed surface-links using biquandle cocycles.

Paper studies knotoid chirality using shadow quandle colorings and invariants.

New cohomology theory reveals $\mathbb{Q}/\mathbb{Z}$ in group homology.

In CJKLS quandle cohomology is used to produce invariants for particular embeddings of codimension two; 2-cocycles give to invariants for (classical) knots and 3-cocycles give rise to invariants for knotted surfaces. This is done by way of a notion of coloring of a diagram. Also, these invariants have the form of state…

We propose a simple method to produce quandle cocycles from group cocycles, as a modification of Inoue-Kabaya chain map. We further show that, in respect to "universal central extended quandles", the chain map induces an isomorphism between their third homologies. For example, all Mochizuki's quandle 3-cocycles are sho…

Given a shadow biquandle $(B,X)$ composed of a biquandle $B$ and a strongly connected $B$-set $X$, we have a local biquandle structure on $X$. The (co)homology groups of such shadow biquandles are isomorphic to those of the corresponding local biquandles. Moreover, cocycle invariants, of oriented links and oriented sur…

From the four normed division algebras--the real numbers, complex numbers, quaternions and octonions, of dimension k=1, 2, 4 and 8, respectively--a systematic procedure gives a 3-cocycle on the Poincare superalgebra in dimensions k+2=3, 4, 6 and 10, and a 4-cocycle on the Poincare superalgebra in dimensions k+3=4, 5, 7…

Let K be the space of long j-knots in R^n. In this paper we introduce a graph complex D and a linear map I from D to the de Rham complex of K via configuration space integral, and prove that (1) when both n>j>=3 are odd, the map I is a cochain map if restricted to graphs with at most one loop component, (2) when n-j>=2…

Novel Lie-superalgebraic description of superstring dynamics.

Study of quandle coloring quivers with dihedral quandles.

Starting from the four normed division algebras - the real numbers, complex numbers, quaternions and octonions - a systematic procedure gives a 3-cocycle on the Poincare Lie superalgebra in dimensions 3, 4, 6 and 10. A related procedure gives a 4-cocycle on the Poincare Lie superalgebra in dimensions 4, 5, 7 and 11. In…

We show that any twisted Dijkgraaf-Witten representation of a mapping class group of an orientable, compact surface with boundary has finite image. This generalizes work of Etingof, Rowell and Witherspoon showing that the braid group images are finite. In particular, our result answers their question regarding finitene…

Quantum cocycle invariants derived from Yang-Baxter cohomology.

Abstract sketches historical development of Lie brackets, crossed modules, and Lie-Rinehart algebras.

A new gauge principle for string models emerges from groupoid symmetries.