arXiv research

Graph homomorphism numbers embed graphs for classification.

Finite subgraphs in flip graphs ensure unique surface embeddings.

A graph homomorphism is a map between two graphs that preserves adjacency relations. We consider the problem of sampling a random graph homomorphism from a graph into a large network. We propose two complementary MCMC algorithms for sampling random graph homomorphisms and establish bounds on their mixing times and the …

Study big mapping class groups and their co-Hopfian property, finding new examples and proving injective homomorphisms results.

When each data point is a large graph, graph statistics such as densities of certain subgraphs (motifs) can be used as feature vectors for machine learning. While intuitive, motif counts are expensive to compute and difficult to work with theoretically. Via graphon theory, we give an explicit quantitative bound for the…

It had been known since old times [MO, Da] that there exists a universal finite type invariant ("an expansion") Z^{old} for Knotted Trivalent Graphs (KTGs), and that it can be chosen to intertwine between some of the standard operations on KTGs and their chord-diagrammatic counterparts (so that relative to those operat…

The paper extends a knot invariant to graphs and connects it to homology cylinders.

The paper classifies dense conjugacy classes in mapping class groups of locally finite graphs.

Two references added and the introduction slightly expanded. We show that the tree-level part of a recent theory of invariants of 3-manifolds (due, independently, to Goussarov and Habiro) is essentially given by classical algebraic topology in terms of the Johnson homomorphism and Massey products, for arbitrary 3-manif…

Mixes higher-order simplicial complexes for data augmentation.

For S a compact connected oriented surface, we consider homology cylinders over S: these are homology cobordisms with an extra homological triviality condition. When considered up to Y_2-equivalence, which is a surgery equivalence relation arising from the Goussarov-Habiro theory, homology cylinders form an Abelian gro…

We study the chromatic number of the curve graph of a surface. We show that the chromatic number grows like k log k for the graph of separating curves on a surface of Euler characteristic -k. We also show that the graph of curves that represent a fixed non-zero homology class is uniquely t-colorable, where t denotes it…

We present a simple proof for the universality of invariant and equivariant tensorized graph neural networks. Our approach considers a restricted intermediate hypothetical model named Graph Homomorphism Model to reach the universality conclusions including an open case for higher-order output. We find that our proposed…

Stable algebraic filters improve neural network performance.

Characterizes a general range decreasing group homomorphism.

An orientation is defined on a family of curve graphs on which the Torelli group acts. It is shown that the resulting signed stable length of an element of the Torelli group is a cohomology class. This cohomology class is half the dual of the contraction of the Johnson homomorphism, the socalled "Chillingworth class".

We extend certain homomorphisms defined on the higher Torelli subgroups of the mapping class group to crossed homomorphisms defined on the entire mapping class group. In particular, for every $k\geq 2$, we construct a crossed homomorphism $ε_k$ which extends Morita's homomorphism $\tilde τ_k$ to the entire mapping clas…

Two crossing homomorphisms on braid groups are shown to be equivalent.

The study classifies homomorphisms from mapping class groups using finite subgroups.

Study of Chebyshev-Frobenius homomorphism in 3-manifold skein modules.

Bestvina-Brady groups arise as kernels of length homomorphisms from right-angled Artin groups G_\G to the integers. Under some connectivity assumptions on the flag complex Δ_\G, we compute several algebraic invariants of such a group N_\G, directly from the underlying graph \G. As an application, we give examples of Be…

We show Péter Csorba's conjecture that the graph homomorphism complex Hom(C_5,K_{n+2}) is homeomorphic to a Stiefel manifold, the space of unit tangent vectors to the n-dimensional sphere. For this a general tool is developed that allows to replace the complexes Hom(G, K_n) by smaller complexes that are homeomorphic to…

New conditions for weighted composition operators in group homomorphisms.

We give a group theoretic characterization of geodesics with superlinear divergence in the Cayley graph of a right-angled Artin group A(G) with connected defining graph G. We use this to determine when two points in an asymptotic cone of A(G) are separated by a cut-point. As an application, we show that if G does not d…

Study homomorphisms from groups to 3-manifold fundamental groups.

New homomorphism from Khovanov homology for knot concordance.

We introduce the notion of tight homomorphism into a locally compact group with nonvanishing bounded cohomology and study these homomorphisms in detail when the target is a Lie group of Hermitian type. Tight homomorphisms between Lie groups of Hermitian type give rise to tight totally geodesic maps of Hermitian symmetr…

A chord index homomorphism for knots in thickened surfaces is constructed.

Study classifies biharmonic and harmonic homomorphisms between specific Lie groups.

Study on Euler class and flux homomorphisms for non-orientable surfaces.

We examine functorial and homotopy properties of the exotic characteristic homomorphism in the category of Lie algebroids which was lastly obtained by the authors in [4]. This homomorphism depends on a triple (A,B,$\nabla$) where B $\subset$ A are regular Lie algebroids, both over the same regular foliated manifold (M,…

We define an infinite family of linearly independent, integer-valued smooth concordance homomorphisms. Our homomorphisms are explicitly computable and rely on local equivalence classes of knot Floer complexes over the ring $\mathbb{F}[U, V]/(UV=0)$. We compare our invariants to other concordance homomorphisms coming fr…

A new homomorphism connects group actions on circles to Euler classes.

We extend each higher Johnson homomorphism to a crossed homomorphism from the automorphism group of a finite-rank free group to a finite-rank abelian group. We also extend each Morita homomorphism to a crossed homomorphism from the mapping class group of once-bounded surface to a finite-rank abelian group. This improve…

We consider bundle homomorphisms between tangent distributions and vector bundles of the same rank. We study the conditions for fundamental singularities when the bundle homomorphism is induced from a Morin map. When the tangent distribution is the contact structure, we characterize singularities of the bundle homomorp…

We propose an approach to study non-Abelian Iwasawa theory, using the idea of Johnson homomorphisms in low dimensional topology. We introduce arithmetic analogues of Johnson homomorphisms/maps, called the p-Johnson homomorphisms/maps, associated to the Zassenhaus filtration of a pro-p Galois group over a Z_p-extension …

Satellite operations with winding number ≠ 1 are not homomorphisms.

Homomorphism from braid groups to Steinberg groups defined.

Let $G$ be a finite group with symmetric generating set $S$, and let $c = \max_{R > 0} |B(2R)|/|B(R)|$ be the doubling constant of the corresponding Cayley graph, where $B(R)$ denotes an $R$-ball in the word-metric with respect to $S$. We show that the multiplicity of the $k$th eigenvalue of the Laplacian on the Cayley…

Johnson and Livingston have characterized peripheral structures in homomorphs of knot groups. We extend their approach to the case of links. The main result is an algebraic characterization of all possible peripheral structures in certain homomorphic images of link groups.

The paper studies symmetries in quandles and their relative versions.

If phi: G-->G' is a surjective homomorphism, we prove that the twisted Alexander polynomial of G is divisible by the twisted Alexander polynomial of G'. As an application, we show non-existence of surjective homomorphism between certain knot groups.

We study the outer automorphism group of a right-angled Artin group $A_Γ$ with finite defining graph $Γ$. We construct a subnormal series for $Out(A_Γ)$ such that each consecutive quotient is either finite, free-abelian, $GL(n,\mathbb{Z})$, or a Fouxe-Rabinovitch group. The last two types act respectively on a symmetri…

Study proves equality of LS-category and cohomological dimension for specific group homomorphisms.

Study de Rham homomorphism for Lipschitz cohomologies on metric simplicial complexes.

New homomorphism from Khovanov homology gives slice genus bounds.

Let K be the kernel of an epimorphism G -> Z, where G is a finitely presented group. If K has infinitely many subgroups of index 2, 3, or 4, then it has uncountably many. Moreover, if K is the commutator subgroup of a classical knot group G, then any homomorphism from K onto the symmetric group S_2 lifts to a homomorph…

Johnson has defined a surjective homomorphism from the Torelli subgroup of the mapping class group of the surface of genus $g$ with one boundary component to $\wedge^3 H$, the third exterior product of the homology of the surface. Morita then extended Johnson's homomorphism to a homomorphism from the entire mapping cla…