130,973 research outputs found
Topological Graph Polynomials in Colored Group Field Theory
In this paper we analyze the open Feynman graphs of the Colored Group Field
Theory introduced in [arXiv:0907.2582]. We define the boundary graph
\cG_{\partial} of an open graph \cG and prove it is a cellular complex.
Using this structure we generalize the topological (Bollobas-Riordan) Tutte
polynomials associated to (ribbon) graphs to topological polynomials adapted to
Colored Group Field Theory graphs in arbitrary dimension
Graph Theory Data for Topological Quantum Chemistry
Topological phases of noninteracting particles are distinguished by global
properties of their band structure and eigenfunctions in momentum space. On the
other hand, group theory as conventionally applied to solid-state physics
focuses only on properties which are local (at high symmetry points, lines, and
planes) in the Brillouin zone. To bridge this gap, we have previously [B.
Bradlyn et al., Nature 547, 298--305 (2017)] mapped the problem of constructing
global band structures out of local data to a graph construction problem. In
this paper, we provide the explicit data and formulate the necessary algorithms
to produce all topologically distinct graphs. Furthermore, we show how to apply
these algorithms to certain "elementary" band structures highlighted in the
aforementioned reference, and so identified and tabulated all orbital types and
lattices that can give rise to topologically disconnected band structures.
Finally, we show how to use the newly developed BANDREP program on the Bilbao
Crystallographic Server to access the results of our computation.Comment: v1: 29 Pages, 13 Figures. Explains how to access the data presented
in arXiv:1703.02050 v2: Accepted version. References updated, figures
improve
Twisty itsy bitsy topological field theory
We extend the topological field theory (``itsy bitsy topological field
theory"') of our previous work from mod-2 to twisted coefficients. This
topological field theory is derived from sutured Floer homology but described
purely in terms of surfaces with signed points on their boundary (occupied
surfaces) and curves on those surfaces respecting signs (sutures). It has
information-theoretic (``itsy'') and quantum-field-theoretic (``bitsy'')
aspects. In the process we extend some results of sutured Floer homology,
consider associated ribbon graph structures, and construct explicit admissible
Heegaard decompositions.Comment: 52 pages, 26 figure
Homology for higher-rank graphs and twisted C*-algebras
We introduce a homology theory for k-graphs and explore its fundamental
properties. We establish connections with algebraic topology by showing that
the homology of a k-graph coincides with the homology of its topological
realisation as described by Kaliszewski et al. We exhibit combinatorial
versions of a number of standard topological constructions, and show that they
are compatible, from a homological point of view, with their topological
counterparts. We show how to twist the C*-algebra of a k-graph by a T-valued
2-cocycle and demonstrate that examples include all noncommutative tori. In the
appendices, we construct a cubical set \tilde{Q}(\Lambda) from a k-graph
{\Lambda} and demonstrate that the homology and topological realisation of
{\Lambda} coincide with those of \tilde{Q}(\Lambda) as defined by Grandis.Comment: 33 pages, 9 pictures and one diagram prepared in TiK
Topological minors of cover graphs and dimension
We show that posets of bounded height whose cover graphs exclude a fixed
graph as a topological minor have bounded dimension. This result was already
proven by Walczak. However, our argument is entirely combinatorial and does not
rely on structural decomposition theorems. Given a poset with large dimension
but bounded height, we directly find a large clique subdivision in its cover
graph. Therefore, our proof is accessible to readers not familiar with
topological graph theory, and it allows us to provide explicit upper bounds on
the dimension. With the introduced tools we show a second result that is
supporting a conjectured generalization of the previous result. We prove that
-free posets whose cover graphs exclude a fixed graph as a topological
minor contain only standard examples of size bounded in terms of .Comment: revised versio
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