Edge stabilization in the homology of graph braid groups

An, Byung Hee; Drummond-Cole, Gabriel C.; Knudsen, Ben

doi:10.2140/gt.2020.24.421

Cited by 14 publications

(20 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A subgraph is a subcomplex of a graph. A graph morphism is a finite composition of isomorphisms onto subcomplexes and inverse subdivisions, which we call smoothings — see Figure 1 and [3, section 2.1].…”

Section: Preliminariesmentioning

confidence: 99%

On the second homology of planar graph braid groups

Knudsen

2022

Journal of Topology

Self Cite

View full text Add to dashboard Cite

show abstract

Section: Preliminariesmentioning

confidence: 99%

On the second homology of planar graph braid groups

Knudsen

2022

Journal of Topology

Self Cite

View full text Add to dashboard Cite

show abstract

“…Cell g 19 corresponds to deleted edge e 0 which is incident at the root * , hence it corresponds to the circular move δ. This way, we translate relator from (20) to the following relation involving geometric generators.…”

Section: A Generalisation Of the Yang-baxter Equationmentioning

confidence: 99%

“…For non-abelian quantum statistics, higher homology groups have been used to extract information about classification of flat complex bundles over C n ( ) [17,18]. Homology groups of graph configuration spaces as well as presentations of graph braid groups are also subjects of independent interest in mathematics, see [19][20][21].…”

Section: Introductionmentioning

confidence: 99%

“…Finally, g 4 can be extracted from ∂{e 12 4 , e 13 8 , 10} using previous results and the expression for g 0 derived from ∂{e 14 0 , e 7 2 , 4}. Let us emphasise that although presentation (20) was derived for n = 3, the relator holds for any n. This is because all relators from Morse presentation for n particles can be extended to relators of the Morse presentation for n + 1 particles by subdividing and adding one more particle next to the root * to all relevant critical cells [31].…”

mentioning

confidence: 99%

“…Let us next interpret presentation (20) in terms of geometric generators that correspond to particle moves on 4 . We have the following correspondence…”

mentioning

confidence: 99%

See 2 more Smart Citations

Geometric Presentations of Braid Groups for Particles on a Graph

Maciążek

2021

Commun. Math. Phys.

Self Cite

View full text Add to dashboard Cite

We study geometric presentations of braid groups for particles that are constrained to move on a graph, i.e. a network consisting of nodes and edges. Our proposed set of generators consists of exchanges of pairs of particles on junctions of the graph and of certain circular moves where one particle travels around a simple cycle of the graph. We point out that so defined generators often do not satisfy the braiding relation known from 2D physics. We accomplish a full description of relations between the generators for star graphs where we derive certain quasi-braiding relations. We also describe how graph braid groups depend on the (graph-theoretic) connectivity of the graph. This is done in terms of quotients of graph braid groups where one-particle moves are put to identity. In particular, we show that for 3-connected planar graphs such a quotient reconstructs the well-known planar braid group. For 2-connected graphs this approach leads to generalisations of the Yang–Baxter equation. Our results are of particular relevance for the study of non-abelian anyons on networks showing new possibilities for non-abelian quantum statistics on graphs.

show abstract

Robertson’s conjecture and universal finite generation in the homology of graph braid groups

Knudsen,

Ramos

2024

Sel. Math. New Ser.

View full text Add to dashboard Cite

We formulate a categorification of Robertson’s conjecture analogous to the categorical graph minor conjecture of Miyata–Proudfoot–Ramos. We show that these conjectures imply the existence of a finite list of atomic graphs generating the homology of configuration spaces of graphs—in fixed degree, with a fixed number of particles, under topological embeddings. We explain how the simplest case of our conjecture follows from work of Barter and Proudfoot–Ramos, implying that the category of cographs is Noetherian, a result of potential independent interest.

show abstract

Edge stabilization in the homology of graph braid groups

Cited by 14 publications

References 22 publications

On the second homology of planar graph braid groups

On the second homology of planar graph braid groups

Geometric Presentations of Braid Groups for Particles on a Graph

Robertson’s conjecture and universal finite generation in the homology of graph braid groups

Contact Info

Product

Resources

About