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Byung Hee An

Publications and source records attributed to Byung Hee An.

At least 19 recordsLinked to original sources
arXiv2026

Hierarchical geometry and right-angled Artin groups in graph braid groups

For the unordered discrete configuration space $\mathrm{UD}_n(\mathsfΓ)$ of $n$ particles on a connected finite graph $\mathsfΓ$, we construct an explicit factor system on its universal cover. Its factors are encoded by legal pairs, namely subgraphs equipped with particle distributions. The nesting, orthogonality, and product regions in the resulting hierarchically hyperbolic group (HHG) structure admit explicit descriptions in terms of configuration-space geometry, and we show that this structure satisfies the additional properties needed for constructing and obstructing subgroups isomorphic to right-angled Artin groups (RAAGs). Using sufficiently subdivided models, we apply this hierarchy to graph braid groups. We give a finite combinatorial formula for the maximal rank of a free abelian subgroup and show that every RAAG occurs as an undistorted subgroup of some graph braid group. For graph $2$-braid groups, we obtain stronger restrictions: every RAAG subgroup has bipartite defining graph, and the embedding problem is characterized by an induced-subgraph condition in the expanded core graph of the hierarchy. For the RAAG defined by the four-vertex path, this condition is equivalent to a finite graphical criterion on the underlying graph.

math.GR↗
arXiv2026

Extending Goldberg's Exact Sequence to Braid Groups of Graphs and Simplicial Complexes

For a finite connected simplicial complex $X$, the strand map $ι_\ast$, from $\mathbb{P}_n(X)$ to $\prod_{i=1}^nπ_1(X,x_i^0)$, sends a pure braid to the homotopy classes of its strands. A theorem of Goldberg (1973) computes its kernel when $X$ is a closed surface other than $S^2$ and $\mathbb{RP}^2$: the kernel is the normal closure of the pure braids supported in an embedded disc. We extend this picture to arbitrary finite connected simplicial complexes. Call $X$ $\textit{weakly Goldberg}$ if some contractible subcomplex $X_0\subseteq X$ realises Goldberg's description, $\kerι_\ast=\left\langle \operatorname{im}(\mathbb{P}_n(X_0)\to\mathbb{P}_n(X)) \right\rangle$, and $\textit{Goldberg}$ if $X_0$ can moreover be chosen so that $\mathbb{P}_n(X_0)\to\mathbb{P}_n(X)$ is injective. We prove that the strand map is surjective if and only if $X\not\cong S^1$; that $X$ is weakly Goldberg if and only if its free part is a forest; and that $X$ is Goldberg if and only if it admits an $\textit{admissible tree}$ -- a maximal tree of a scaffold, compatible with the boundary and interior types of the attachments of the free part to the thick components. We also classify the complexes for which the kernel is trivial, settle the exceptional surfaces $S^2$ and $\mathbb{RP}^2$, and obtain complete answers for manifolds and for graphs. The main tools are a graph-of-spaces decomposition of the configuration space at a point of $X$ and a resolution procedure reducing an arbitrary complex to a simple model.

math.AT↗
arXiv2026

On the large-scale geometry of graph braid groups via cubical structures

We study the large-scale geometry of graph braid groups $\mathbb{B}_n(\mathsfΓ)$, viewed as the fundamental groups of discrete configuration spaces $UD_n(\mathsfΓ)$, which are special cube complexes in the sense of Haglund--Wise. Exploiting this cubical structure, we relate hyperbolicity, undistorted surface subgroups, and group-theoretic decompositions. As a consequence, we obtain a complete classification of when $\mathbb{B}_n(\mathsfΓ)$ is quasi-isometric to a free group via a purely geometric argument independent of discrete Morse theory. We then focus on graph $2$-braid groups. Using maximal product subcomplexes of $UD_2(\mathsfΓ)$ and the intersection complex introduced in \cite{Oh22}, we show that, under natural assumptions, their union captures essential quasi-isometry information about $\mathbb{B}_2(\mathsfΓ)$. As applications, we construct infinitely many graph $2$-braid groups that are quasi-isometric to right-angled Artin groups and infinitely many that are not, extending \cite{Oh22}, and we exhibit new phenomena in relative hyperbolicity.

math.GT↗
arXiv2026

Corrigendum to "Asymptotic homology of graph braid groups"

We correct an error in the paper referred to in the title. Part of the argument is organized as a general method for establishing when (derived) functors factor through a fixed Serre subcategory, which may be of some more general interest.

math.AT↗
arXiv2025

On coshuffle comultiplication on configuration spaces

We introduce a coshuffle comultiplication on the singular chain complex of configuration spaces, and we show that this structure endows the configuration space with the structure of a differential graded coalgebra (DGCoAlg). We then prove that the coshuffle comultiplication is compatible with the external product through a natural commutation relation. As an application, we investigate configuration spaces of graphs and the associated graph braid groups. In particular, for graphs of topological circumference at most 1, we prove that the singular chain complex of the configuration space is formal as a DGCoAlg. Moreover, we obtain a complete classification of the primitivity in the homology of configuration spaces of such graphs.

math.AT↗
arXiv2025

Hilbert polynomials of configuration spaces over graphs of circumference at most 1

The $ k $-configuration space $ B_kΓ$ of a topological space $ Γ$ is the space of sets of $ k $ distinct points in $ Γ$. In this paper, we consider the case where $ Γ$ is a graph of circumference at most $1$. We show that for all $ k\ge0 $, the $ i $-th Betti number of $ B_kΓ$ is given by a polynomial $P_Γ^i(k)$ in $ k $, called the Hilbert polynomial of $ Γ$. We find an expression for the Hilbert polynomial $P_Γ^i(k)$ in terms of those coming from the canonical $1$-bridge decomposition of $ Γ$. We also give a combinatorial description of the coefficients of $P_Γ^i(k)$.

math.GT↗
arXiv2025

Quasi-isometry classification of certain graph $2$-braid groups and its applications

In \cite{Oh22}, the second author defined a complex of groups decomposition of the fundamental group of a finitely generated 2-dimensional special group, called an \emph{intersection complex}, which is a quasi-isometry invariant. In this paper, using the theory of intersection complexes, we classify the class of 2-braid groups over graphs with circumference $\leq 1$ up to quasi-isometry. Moreover, we find a sufficient condition when such a graph 2-braid group is quasi-isometric to a right-angled Artin group or not. Finally, by applying the same method, we also find that there is an algorithm to determine whether two 4-braid groups over trees are quasi-isometric or not.

math.GR↗
arXiv2024

Lagrangian fillings for Legendrian links of finite or affine Dynkin type

We prove that there are at least as many exact embedded Lagrangian fillings as seeds for Legendrian links of finite type $\mathsf{ADE}$ or affine type $\tilde{\mathsf{D}} \tilde{\mathsf{E}}$. We also provide as many Lagrangian fillings with rotational symmetry as seeds of type $\mathsf{B}$, $\mathsf{G}_2$, $\tilde{\mathsf{G}}_2$, $\tilde{\mathsf{B}}$, or $\tilde{\mathsf{C}}_2$, and with conjugation symmetry as seeds of type $\mathsf{F}_4$, $\mathsf{C}$, $\mathsf{E}_6^{(2)}$, $\tilde{\mathsf{F}}_4$, or $\mathsf{A}_5^{(2)}$. These families are the first known Legendrian links with (infinitely many) exact Lagrangian fillings (with symmetry) that exhaust all seeds in the corresponding cluster structures beyond type $\mathsf{A} \mathsf{D}$. Furthermore, we show that the $N$-graph realization of (twice of) Coxeter mutation of type $\tilde{\mathsf{D}} \tilde{\mathsf{E}}$ corresponds to a Legendrian loop of the corresponding Legendrian links. Especially, the loop of type $\tilde{\mathsf{D}}$ coincides with the one considered by Casals and Ng.

math.SG↗
arXiv2022

On the second homology of planar graph braid groups

We show that the second homology of the configuration spaces of a planar graph is generated under the operations of embedding, disjoint union, and edge stabilization by three atomic graphs: the cycle graph with one edge, the star graph with three edges, and the theta graph with four edges. We give an example of a non-planar graph for which this statement is false.

math.AT↗
arXiv2021

Lagrangian fillings for Legendrian links of affine type

We prove that there are at least as many exact embedded Lagrangian fillings as seeds for Legendrian links of affine type $\tilde{\mathsf{D}} \tilde{\mathsf{E}}$. We also provide as many Lagrangian fillings with certain symmetries as seeds of type $\tilde{\mathsf{B}}_n$, $\tilde{\mathsf{F}}_4$, $\tilde{\mathsf{G}}_2$, and $\mathsf{E}_6^{(2)}$. These families are the first known Legendrian links with infinitely many fillings that exhaust all seeds in the corresponding cluster structures. Furthermore, we show that Legendrian realization of Coxeter mutation of type $\tilde{\mathsf{D}}$ corresponds to the Legendrian loop considered by Casals and Ng.

math.SG↗
arXiv2021

On folded cluster patterns of affine type

A cluster algebra is a commutative algebra whose structure is decided by a skew-symmetrizable matrix or a quiver. When a skew-symmetrizable matrix is invariant under an action of a finite group and this action is admissible, the folded cluster algebra is obtained from the original one. Any cluster algebra of non-simply-laced affine type can be obtained by folding a cluster algebra of simply-laced affine type with a specific $G$-action. In this paper, we study the combinatorial properties of quivers in the cluster algebra of affine type. We prove that for any quiver of simply-laced affine type, $G$-invariance and $G$-admissibility are equivalent. This leads us to prove that the set of $G$-invariant seeds forms the folded cluster pattern.

math.CO↗
arXiv2021

Lagrangian fillings for Legendrian links of finite type

We prove that there are at least seeds many exact embedded Lagrangian fillings for Legendrian links of type $\mathsf{ADE}$. We also provide seeds many Lagrangian fillings with certain symmetries for type $\mathsf{BCFG}$. Our main tools are $N$-graphs and the combinatorics of seed patterns of finite type.

math.SG↗
arXiv2020

Universal properties of anyon braiding on one-dimensional wire networks

We demonstrate that anyons on wire networks have fundamentally different braiding properties than anyons in 2D. Our analysis reveals an unexpectedly wide variety of possible non-abelian braiding behaviours on networks. The character of braiding depends on the topological invariant called the connectedness of the network. As one of our most striking consequences, particles on modular networks can change their statistical properties when moving between different modules. However, sufficiently highly connected networks already reproduce braiding properties of 2D systems. Our analysis is fully topological and independent on the physical model of anyons.

quant-ph↗
arXiv2020

Geometric presentations of braid groups for particles on a graph

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.

math-ph↗
arXiv2020

Subdivisional spaces and graph braid groups

We study the problem of computing the homology of the configuration spaces of a finite cell complex $X$. We proceed by viewing $X$, together with its subdivisions, as a subdivisional space--a kind of diagram object in a category of cell complexes. After developing a version of Morse theory for subdivisional spaces, we decompose $X$ and show that the homology of the configuration spaces of $X$ is computed by the derived tensor product of the Morse complexes of the pieces of the decomposition, an analogue of the monoidal excision property of factorization homology. Applying this theory to the configuration spaces of a graph, we recover a cellular chain model due to Świątkowski. Our method of deriving this model enhances it with various convenient functorialities, exact sequences, and module structures, which we exploit in numerous computations, old and new.

math.AT↗
arXiv2019

Augmentations are sheaves for Legendrian graphs

In this article, associated to a (bordered) Legendrian graph, we study and show the equivalence between two categorical Legendrian isotopy invariants: the augmentation category, a unital $A_{\infty}$-category, which lifts the set of augmentations of the associated Chekanov-Eliashberg DGA, and a DG category of constructible sheaves on the front plane, with micro-support at contact infinity controlled by the (bordered) Legendrian graph. In other words, generalizing [21], we prove "augmentations are sheaves" in the singular case.

math.SG↗