Presentations of graph braid groups

Farley, Daniel; Sabalka, Lucas

doi:10.1515/form.2011.086

Cited by 24 publications

(28 citation statements)

References 17 publications

(79 reference statements)

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“…Proposition 4.8 extends the result about 2-point braid groups of graphs with only disjoint cycles [6,Theorem 5.6] to a wider class of graphs including all light planar graphs.…”

Section: Computing Graph Braid Groupssupporting

confidence: 63%

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Computing braid groups of graphs with applications to robot motion planning

Kurlin

2012

Homology, Homotopy and Applications

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An algorithm is designed to write down presentations of graph braid groups. Generators are represented in terms of actual motions of robots moving without collisions on a given connected graph. A key ingredient is a new motion planning algorithm whose complexity is linear in the number of edges and is quadratic in the number of robots. The computing algorithm implies that 2-point braid groups of all light planar graphs have presentations where all relators are commutators.

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“…Proposition 4.8 extends the result about 2-point braid groups of graphs with only disjoint cycles [6,Theorem 5.6] to a wider class of graphs including all light planar graphs.…”

Section: Computing Graph Braid Groupssupporting

confidence: 63%

“…According to [6,Theorem 5.6], the braid groups of planar graphs having only disjoint cycles have presentations where each relator is a commutator, not necessarily a commutator of generators. To demonstrate the power of Algorithm 1.5, this result is extended to a wider class of light planar graphs.…”

Section: Resultsmentioning

confidence: 99%

Computing braid groups of graphs with applications to robot motion planning

Kurlin

2012

Homology, Homotopy and Applications

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“…In [9], Farley and Sabalka conjectured that B 2 Γ is simple-commutator-related for a planar graph Γ and relators are commutators of two words that represent disjoint circuits on the planar graph. In a private correspondence, Abrams conjectured that P 2 Γ is simple-commutator-related for a planar graph Γ.…”

Section: 3mentioning

confidence: 99%

Characteristics of Graph Braid Groups

Park

2012

Discrete Comput Geom

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We give formulae for the first homology of the n-braid group and the pure 2-braid group over a finite graph in terms of graph theoretic invariants. As immediate consequences, a graph is planar if and only if the first homology of the n-braid group over the graph is torsion-free and the conjectures about the first homology of the pure 2-braid groups over graphs in [10] can be verified. We discover more characteristics of graph braid groups: the n-braid group over a planar graph and the pure 2-braid group over any graph have a presentation whose relators are words of commutators, and the 2-braid group and the pure 2-braid group over a planar graph have a presentation whose relators are commutators. The latter was a conjecture in [9] and so we propose a similar conjecture for higher braid indices.2010 Mathematics Subject Classification. Primary 20F36, 20F65, 57M15.

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“…, ar. Some relevant fundamental groups groups are the fundamental group for the n-particle configuration space of R 3 , which is the permutation group Sn, the fundamental group for the n-particle configuration space of R 2 , which is the braid group on n strands Brn, the fundamental group for the n-particle configuration space of a graph, which is a graph braid group [28,29].…”

Section: Methodsmentioning

confidence: 99%

Non-abelian Quantum Statistics on Graphs

Maciążek

Sawicki

2019

Commun. Math. Phys.

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We show that non-abelian quantum statistics can be studied using certain topological invariants which are the homology groups of configuration spaces. In particular, we formulate a general framework for describing quantum statistics of particles constrained to move in a topological space X. The framework involves a study of isomorphism classes of flat complex vector bundles over the configuration space of X which can be achieved by determining its homology groups. We apply this methodology for configuration spaces of graphs. As a conclusion, we provide families of graphs which are good candidates for studying simple effective models of anyon dynamics as well as models of non-abelian anyons on networks that are used in quantum computing. These conclusions are based on our solution of the so-called universal presentation problem for homology groups of graph configuration spaces for certain families of graphs.

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Presentations of graph braid groups

Cited by 24 publications

References 17 publications

Computing braid groups of graphs with applications to robot motion planning

Computing braid groups of graphs with applications to robot motion planning

Characteristics of Graph Braid Groups

Non-abelian Quantum Statistics on Graphs

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