Aspherical Word Labeled Oriented Graphs and Cyclically Presented Groups

A {\em word labeled oriented graph} (WLOG) is an oriented graph $\cal G$ on vertices $X=\{ x_1,\ldots ,x_k\}$, where each oriented edge is labeled by a word in $X^{\pm1}$. WLOGs give rise to presentations which generalize Wirtinger presentations of knots. WLOG presentations, where the underlying graph is a tree are of central importance in view of Whitehead's Asphericity Conjecture. We present a class of aspherical world labeled oriented graphs. This class can be used to produce highly non-injective aspherical labeled oriented trees and also aspherical cyclically presented groups.Comment: 7 pages, 3 figure

Ribbon 2-knot groups of Coxeter type

Wirtinger presentations of deficiency 1 appear in the context of knots, long virtual knots, and ribbon 2-knots. They are encoded by (word) labeled oriented trees and, for that reason, are also called LOT presentations. These presentations are a well known and important testing ground for the validity (or failure) of Whitehead's asphericity conjecture. In this paper we define LOTs of Coxeter type and show that for every given $n$ there exists a (prime) LOT of Coxeter type with group of rank $n$. We also show that label separated Coxeter LOTs are aspherical

On the \u3ci\u3eK\u3c/i\u3e-Theory and Homotopy Theory of the Klein Bottle Group

We construct infinitely many chain homotopically distinct algebraic 2-complexes for the Klein bottle group and give various topological applications. We compare our examples to other examples in the literature and address the question of geometric realizability

The Local Structure of Injective LOT-Complexes

Labeled oriented trees, LOT's, encode spines of ribbon discs in the 4-ball and ribbon 2-knots in the 4-sphere. The unresolved asphericity question for these spines is a major test case for Whitehead's asphericity conjecture. In this paper we give a complete description of the link of a reduced injective LOT complex. An important case is the following: If $\Gamma$ is a reduced injective LOT that does not contain boundary reduced sub-LOTs, then $lk(K(\Gamma))$ is a bi-forest. As a consequence $K(\Gamma)$ is aspherical, in fact DR, and its fundamental group is locally indicable. We also show that a general injective LOT complex is aspherical. Some of our results have already appeared in print over the last two decades and are collected here

The Σ2-conjecture for metabelian groups: the general case

AbstractThe Bieri–Neumann–Strebel invariant Σm(G) of a group G is a certain subset of a sphere that contains information about finiteness properties of subgroups of G. In case of a metabelian group G the set Σ1(G) completely characterizes finite presentability and it is conjectured that it also contains complete information about the higher finiteness properties (FPm-conjecture). The Σm-conjecture states how the higher invariants are obtained from Σ1(G). In this paper we prove the Σ2-conjecture

Gluon-Induced Weak Boson Fusion

The gluon-gluon induced terms for Higgs production through weak boson fusion (WBF) are computed. Formally, these are of NNLO in the strong coupling constant. This is the lowest order at which non-zero color exchange occurs between the scattering quarks, leading to a color field and thus additional hadronic activity between the outgoing jets. Using a minimal set of cuts, the numerical impact of these terms is at the percent level with respect to the NLO rate for weak boson fusion. Applying the so-called WBF cuts leads to an even stronger suppression, so that we do not expect a significant deterioration of the WFB signal by these color exchange effects.Comment: 9 pages, 8 figures (21 included ps- and eps-files