2,203 research outputs found
Graph braid groups and right-angled Artin groups
We give a necessary and sufficient condition for a graph to have a
right-angled Artin group as its braid group for braid index . In order
to have the necessity part, graphs are organized into small classes so that one
of homological or cohomological characteristics of right-angled Artin groups
can be applied. Finally we show that a given graph is planar iff the first
homology of its 2-braid group is torsion-free and leave the corresponding
statement for -braid groups as a conjecture along with few other conjectures
about graphs whose braid groups of index are right-angled Artin groups.Comment: 52 pages, 30 figure
An introduction to right-angled Artin groups
Recently, right-angled Artin groups have attracted much attention in
geometric group theory. They have a rich structure of subgroups and nice
algorithmic properties, and they give rise to cubical complexes with a variety
of applications. This survey article is meant to introduce readers to these
groups and to give an overview of the relevant literature
Graph 4-braid groups and Massey products
We first show that the braid group over a graph topologically containing no
-shape subgraph has a presentation related only by commutators. Then
using discrete Morse theory and triple Massey products, we prove that a graph
topologically contains none of four prescribed graphs if and only if its
4-braid groups is a right-angled Artin group.Comment: 23 pages, 4 figure
New Categorifications of the Chromatic and the Dichromatic Polynomials for Graphs
In this paper, for each graph , we def\mbox{}ine a chain complex of graded
modules over the ring of polynomials, whose graded Euler characteristic is
equal to the chromatic polynomial of . Furthermore, we def\mbox{}ine a chain
complex of doubly-graded modules, whose (doubly) graded Euler characteristic is
equal to the dichromatic polynomial of . Both constructions use Koszul
complexes, and are similar to the new Khovanov-Rozansky
categorif\mbox{}ications of HOMFLYPT polynomial. We also give simplif\mbox{}ied
def\mbox{}inition of this triply-graded link homology theory.Comment: 15 pages, added Section
Torsion in Khovanov homology of semi-adequate links
The goal of this paper is to address A. Shumakovitch's conjecture about the
existence of -torsion in Khovanov link homology. We analyze torsion in
Khovanov homology of semi-adequate links via chromatic cohomology for graphs
which provides a link between the link homology and well-developed theory of
Hochschild homology. In particular, we obtain explicit formulae for torsion and
prove that Khovanov homology of semi-adequate links contains -torsion if
the corresponding Tait-type graph has a cycle of length at least 3.
Computations show that torsion of odd order exists but there is no general
theory to support these observations. We conjecture that the existence of
torsion is related to the braid index.Comment: 23 pages, 7 figure
Random walks on hyperbolic groups and their Riemann surfaces
We investigate invariants for random elements of different hyperbolic groups.
We provide a method, using Cayley graphs of groups, to compute the probability
distribution of the minimal length of a random word, and explicitly compute the
drift in different cases, including the braid group . We also compute in
this case the return probability. The action of these groups on the hyperbolic
plane is investigated, and the distribution of a geometric invariant, the
hyperbolic distance, is given. These two invariants are shown to be related by
a closed formula.Comment: 29 pages, 8 figure
Magic-State Functional Units: Mapping and Scheduling Multi-Level Distillation Circuits for Fault-Tolerant Quantum Architectures
Quantum computers have recently made great strides and are on a long-term
path towards useful fault-tolerant computation. A dominant overhead in
fault-tolerant quantum computation is the production of high-fidelity encoded
qubits, called magic states, which enable reliable error-corrected computation.
We present the first detailed designs of hardware functional units that
implement space-time optimized magic-state factories for surface code
error-corrected machines. Interactions among distant qubits require surface
code braids (physical pathways on chip) which must be routed. Magic-state
factories are circuits comprised of a complex set of braids that is more
difficult to route than quantum circuits considered in previous work [1]. This
paper explores the impact of scheduling techniques, such as gate reordering and
qubit renaming, and we propose two novel mapping techniques: braid repulsion
and dipole moment braid rotation. We combine these techniques with graph
partitioning and community detection algorithms, and further introduce a
stitching algorithm for mapping subgraphs onto a physical machine. Our results
show a factor of 5.64 reduction in space-time volume compared to the best-known
previous designs for magic-state factories.Comment: 13 pages, 10 figure
On the loxodromic actions of Artin-Tits groups
Artin-Tits groups act on a certain delta-hyperbolic complex, called the
"additional length complex". For an element of the group, acting loxodromically
on this complex is a property analogous to the property of being pseudo-Anosov
for elements of mapping class groups. By analogy with a well-known conjecture
about mapping class groups, we conjecture that "most" elements of Artin-Tits
groups act loxodromically. More precisely, in the Cayley graph of a subgroup
of an Artin-Tits group, the proportion of loxodromically acting elements in
a ball of large radius should tend to one as the radius tends to infinity. In
this paper, we give a condition guaranteeing that this proportion stays away
from zero. This condition is satisfied e.g. for Artin-Tits groups of spherical
type, their pure subgroups and some of their commutator subgroups.Comment: 9 pages, 2 figures, 1 tabl
Coxeter group actions on the complement of hyperplanes and special involutions
We consider both standard and twisted action of a (real) Coxeter group G on
the complement M_G to the complexified reflection hyperplanes by combining the
reflections with complex conjugation. We introduce a natural geometric class of
special involutions in G and give explicit formulae which describe both actions
on the total cohomology H(M_G,C) in terms of these involutions. As a corollary
we prove that the corresponding twisted representation is regular only for the
symmetric group S_n, the Weyl groups of type D_{2m+1}, E_6 and dihedral groups
I_2 (2k+1) and that the standard action has no anti-invariants. We discuss also
the relations with the cohomology of generalised braid groups.Comment: 11 page
Matrix factorizations and link homology II
To a presentation of an oriented link as the closure of a braid we assign a
complex of bigraded vector spaces. The Euler characteristic of this complex
(and of its triply-graded cohomology groups) is the HOMFLYPT polynomial of the
link. We show that the dimension of each cohomology group is a link invariant.Comment: 37 pages, 20 figures; version 2 corrects an inaccuracy in the proof
of Proposition
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