
Ruling invariants for Legendrian graphs, to appear in Journal of Symplectic Geometry.
arXiv
Abstract
(with Youngjin Bae(INU) and Tamás Kálmán(TIT))
We define ruling invariants for evenvalence Legendrian graphs in standard contact threespace. We prove that rulings exist if and only if the DGA of the graph, introduced by the first two authors, has an augmentation. We set up the usual ruling polynomials for various notions of gradedness and prove that if the graph is fourvalent, then the ungraded ruling polynomial appears in KauffmanVogel's graph version of the Kauffman polynomial. Our ruling invariants are compatible with certain vertexidentifying operations as well as vertical cuts and gluings of front diagrams. We also show that Leverson's definition of a ruling of a Legendrian link in a connected sum of $S^1\times S^2$'s can be seen as a special case of ours.

Geometric presentations of graph braid groups, to appear in Communications in Mathematical Physics.
arXiv
Abstract
(with Tomasz Maciążek(University of Bristol))
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 quasibraiding relations. We also describe how graph braid groups depend on the (graphtheoretic) connectivity of the graph. This is done in terms of quotients of graph braid groups where oneparticle moves are put to identity. In particular, we show that for $3$connected planar graphs such a quotient reconstructs the wellknown planar braid group. For $2$connected graphs this approach leads to generalisations of the YangBaxter equation. Our results are of particular relevance for the study of nonabelian anyons on networks showing new possibilities for nonabelian quantum statistics on graphs.

Augmentations and ruling polynomials for Legendrian graphs, to appear in Algebraic & Geometric Topology.
arXiv
Abstract
(with Youngjin Bae(INU) and Tao Su(ENS))
In this article, associated to a (bordered) Legendrian graph, we study and show the equivalence between two Legendrian isotopy invariants: augmentation number via pointcounting over a finite field, for the augmentation variety of the associated ChekanovEliashberg differential graded algebra, and ruling polynomial via combinatorics of the decompositions of the associated front projection.

Universal properties of anyon braiding on onedimensional wire networks, Physical Review B, 102 (2020), Issue 20, 201407(R).
DOI
arXiv
Abstract
(with Tomasz Maciążek(University of Bristol))
We demonsrtrate that anyons on wire networks have fundamentally different braiding properties than anyons in 2D. Our analysis reveals an unexpectedly wide variety of possible nonabelian 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.

A ChekanovEliashberg algebra for Legendrian graphs, Journal of Topology, 13 (2020), no. 2, 777–869.
DOI
arXiv
Abstract
(with Youngjin Bae(KIAS))
We define a differential graded algebra for Legendrian graphs and tangles in the standard contact Euclidean three space. This invariant is defined combinatorially by using ideas from Legendrian contact homology. The construction is distinguished from other versions of Legendrian contact algebra by the vertices of Legendrian graphs. A set of countably many generators and a generalized notion of equivalence are introduced for invariance. We show a van Kampen type theorem for the differential graded algebras under the tangle replacement. Our construction recovers many known algebraic constructions of Legendrian links via suitable operations at the vertices.

Edge stabilization in the homology of graph braid groups, Geometry and Topology, 24 (2020), no. 1, 421–469.
DOI
arXiv
Abstract
(with Gabriel C. DrummondCole(IBSCGP) and Ben Knudsen(Harvard University))
We introduce a novel type of stabilization map on the configuration spaces of a graph, which increases the number of particles occupying an edge. There is an induced action on homology by the polynomial ring generated by the set of edges, and we show that this homology module is finitely generated. An analogue of classical homological and representation stability for manifolds, this result implies eventual polynomial growth of Betti numbers. We calculate the exact degree of this polynomial, in particular verifying an upper bound conjectured by Ramos. Because the action arises from a family of continuous maps, it lifts to an action at the level of singular chains, which contains strictly more information than the homology level action. We show that the resulting differential graded module is almost never formal over the ring of edges.

Subdivisional spaces and graph braid groups, Doc. Math., 24 (2019), 1513–1583.
DOI
arXiv
Abstract
(with Gabriel C. DrummondCole(IBSCGP) and Ben Knudsen(Harvard University))
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 spacea 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.

On the Chern numbers for pseudofree circle actions, J. Symplectic Geom., 17 (2019), no. 1, 1–40.
DOI
arXiv
Abstract
(with Yunhyung Cho(SKKU))
Let $(M,\psi)$ be a $(2n+1)$dimensional oriented closed manifold with a pseudofree $S^1$action
$\psi : S^1 \times M \rightarrow M$.
We first define a \textit{local data} $\mathcal{L}(M,\psi)$ of the action $\psi$ which consists of
pairs $(C, (p(C) ; \overrightarrow{q}(C)))$ where $C$ is an exceptional orbit, $p(C)$ is the order of isotropy subgroup of $C$, and $\overrightarrow{q}(C) \in (\mathbb{Z}_{p(C)}^{\times})^n$ is a vector whose entries are the weights of the slice representation of $C$.
In this paper, we give an explicit formula of the Chern number $\langle c_1(E)^n, [M/S^1] \rangle$ modulo $\mathbb{Z}$ in terms of
the local data, where $E = M \times_{S^1} \mathbb{C}$ is the associated complex line orbibundle
over $M/S^1$.
Also, we illustrate several applications to various problems arising in equivariant symplectic topology.

Legendrian singular links and singular connected sums, J. Symplectic Geom., 16 (2018), no. 4, 885–930.
DOI
arXiv
Abstract
(with Youngjin Bae(RIMS) and Seonhwa Kim(IBSCGP))
We study Legendrian singular links up to contact isotopy. Using a special property of the singular points, we define the singular connected sum of Legendrian singular links. This concept is a generalization of the connected sum and can be interpreted as a tangle replacement, which provides a way to classify Legendrian singular links. Moreover, we investigate several phenomena only occur in the Legendrian setup.

Grid diagrams for singular links, J. Knot Theory Ramification, 27 (2018), no. 4, 1850023, 43pp.
DOI
arXiv
Abstract
(with Hwa Jeong Lee(DGIST))
In this paper, we define the set of singular grid diagrams $\mathcal{SG}$ which provides a unified description for singular links, singular Legendrian links, singular transverse links, and singular braids. We also classify the complete set of all equivalence relations on $\mathcal{SG}$ which induce the bijection onto each singular object. This is an extension of the known result of Ng–Thurston [Grid diagrams, braids, and contact geometry, in Proc. Gökova GeometryTopology Conf. 2008, Gökova Geometry/Topology Conference (GGT), Gökova, 2009, pp. 120–136] for nonsingular links and braids.

On the $f$vectors of GelfandCetlin polytopes, European J. Combin., 67 (2018), 61–77.
DOI
arXiv
Abstract
(with Yunhyung Cho(SKKU) and Jang Soo Kim(SKKU))
A Gelfand–Cetlin polytope is a convex polytope obtained as an image of certain completely integrable system on a partial flag variety. In this paper, we give an equivalent description of the face structure of a GCpolytope in terms of so called the face structure of a ladder diagram. Using our description, we obtain a partial differential equation whose solution is the exponential generating function of $f$vectors of GCpolytopes. This solves the open problem (2) posed by Gusev et al. (2013).

On the structure of braid groups on complexes, Topology Appl., 226 (2017), 86–119.
DOI
arXiv
Abstract
(with Hyowon Park)
We consider the braid group $\mathbf{B}_n(X)$ on a finite simplicial complex $X$, which is a generalization of those on both manifolds and graphs that have been studied already by many authors. We figure out the relationships between geometric decompositions for $X$ and their effects on the braid groups.
As applications, we give complete criteria for both the surface embeddability and planarity for $X$, which are the torsionfreeness of the braid group $\mathbf{B}_n(X)$ and its abelianization $H_1(\mathbf{B}_n(X))$, respectively.

A criterion for the Legendrian simplicity of the connected sum, Topology Appl., 204 (2016), 175–184.
DOI
arXiv
Abstract
In this paper, we provide the necessary and sufficient conditions for the connected sum of knots in $S^3$ to be Legendrian simple.

Automorphisms of braid groups on orientable surfaces, J. Knot Theory Ramification, 25 (2016), no. 5, 1650022, 32pp.
DOI
arXiv
Abstract
In this paper we compute the automorphism groups $\operatorname{Aut}(\mathbf{P}_n(\Sigma))$ and $\operatorname{Aut}(\mathbf{B}_n(\Sigma))$ of braid groups $\mathbf{P}_n(\Sigma)$ and $\mathbf{B}_n(\Sigma)$ on every orientable surface $\Sigma$, which are isomorphic to group extensions of the extended mapping class group $\mathcal{M}^*_n(\Sigma)$ by the transvection subgroup except for a few cases.
We also prove that $\mathbf{P}_n(\Sigma)$ is always a characteristic subgroup of $\mathbf{B}_n(\Sigma)$ unless $\Sigma$ is a twicepunctured sphere and $n=2$.

A family of pseudoAnosov braids with large conjugacy invariant sets, J. Knot Theory Ramification, 22 (2013), no. 6, 1350025, 20pp.
DOI
arXiv
Abstract
(with Ki Hyoung Ko(KAIST))
We show that there is a family of pseudoAnosov braids independently parametrized by the braid index and the (canonical) length whose smallest conjugacy invariant sets grow exponentially in the braid index and linearly in the length.

A family of representations of braid groups on surfaces, Pacific J. Math., 247 (2010), no. 2, 257–282.
DOI
arXiv
Abstract
(with Ki Hyoung Ko(KAIST))
We propose a family of homological representations of the braid groups on surfaces. This family extends linear representations of the braid groups on a disc, such as the Burau representation and the LawrenceKrammerBigelow representation.