IA Scholar Query: Isomorphism for Graphs Embeddable on the Projective Plane
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgTue, 20 Sep 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440Simplicial approximation to CW complexes in practice
https://scholar.archive.org/work/jcypnswcf5hgjlw765ea6c2xom
We describe an algorithm that takes as an input a CW complex and returns a simplicial complex of the same homotopy type. This algorithm, although well-known in the literature, requires some work to make it computationally tractable. We pay close attention to weak simplicial approximation, which we implement for two subdivisions methods: generalized barycentric and generalized edgewise subdivisions. We also propose a new subdivision process, based on Delaunay complexes. In order to facilitate the computation of a simplicial approximation, we introduce a simplification step, based on edge contractions. We define a new version of simplicial mapping cone, which requires less simplices. Last, we illustrate the algorithm with the real projective spaces, the 3-dimensional lens spaces and the Grassmannian of 2-planes in ℝ^4. As applications of these results, we estimate the discrete Lusternik-Schnirelmann category of our lens spaces, and we compute the persistent Stiefel-Whitney classes of a dataset of plane bundles.Raphaël Tinarragework_jcypnswcf5hgjlw765ea6c2xomTue, 20 Sep 2022 00:00:00 GMTConformal field theory from lattice fermions
https://scholar.archive.org/work/paiceamvtbhplclnowhin3apde
We provide a rigorous lattice approximation of conformal field theories given in terms of lattice fermions in 1+1-dimensions, focussing on free fermion models and Wess-Zumino-Witten models. To this end, we utilize a recently introduced operator-algebraic framework for Wilson-Kadanoff renormalization. In this setting, we prove the convergence of the approximation of the Virasoro generators by the Koo-Saleur formula. From this, we deduce the convergence of lattice approximations of conformal correlation functions to their continuum limit. In addition, we show how these results lead to explicit error estimates pertaining to the quantum simulation of conformal field theories.Tobias J. Osborne, Alexander Stottmeisterwork_paiceamvtbhplclnowhin3apdeThu, 15 Sep 2022 00:00:00 GMTGeometric Manin's Conjecture for Fano 3-Folds
https://scholar.archive.org/work/hwms6q6wfjhrnclguhkdc2t6fu
We classify families of irreducible, nef rational curves on general members of all 88 families of smooth Fano threefolds of Picard rank at least two. This proves Geometric Manin's Conjecture for general members of all 88 families, and for arbitrary members of 75 families.Andrew Burke, Eric Jovinellywork_hwms6q6wfjhrnclguhkdc2t6fuMon, 12 Sep 2022 00:00:00 GMTHereditarily indecomposable continua as generic mathematical structures
https://scholar.archive.org/work/sn72ryh7zfga5evmouk42e7h4a
We characterize the pseudo-arc as well as P-adic pseudo-solenoids (for a set of primes P) as generic structures, arising from a natural game in which two players alternate in building an inverse sequence of surjections. The second player wins if the limit of this sequence is homeomorphic to a concrete (fixed in advance) space, called generic whenever the second player has a winning strategy. For this aim, we develop a new approximate Fra\"iss\'e theory, in order to realize the above-mentioned objects (the pseudo-arc and the pseudo-solenoids) as Fra\"iss\'e limits. Our framework extends the discrete Fra\"iss\'e theory, both classical and projective, and is also suitable for working directly with continuous maps on metrizable compacta. We show, in particular, that, when playing with continuous surjections between non-degenerate Peano continua, the pseudo-arc is always generic. The universal pseudo-solenoid appears to be generic over all surjections between circle-like continua.Adam Bartoš, Wiesław Kubiśwork_sn72ryh7zfga5evmouk42e7h4aFri, 02 Sep 2022 00:00:00 GMTHanani-Tutte for approximating maps of graphs
https://scholar.archive.org/work/ylinuk5p7zf4xejevb4ab2olxq
We resolve in the affirmative conjectures of Repovs and A. Skopenkov (1998), and M. Skopenkov (2003) generalizing the classical Hanani-Tutte theorem to the setting of approximating maps of graphs on 2-dimensional surfaces by embeddings. Our proof of this result is constructive and almost immediately implies an efficient algorithm for testing if a given piecewise linear map of a graph in a surface is approximable by an embedding. More precisely, an instance of this problem consists of (i) a graph G whose vertices are partitioned into clusters and whose inter-cluster edges are partitioned into bundles, and (ii) a region R of a 2-dimensional compact surface M given as the union of a set of pairwise disjoint discs corresponding to the clusters and a set of pairwise non-intersecting "pipes" corresponding to the bundles, connecting certain pairs of these discs. We are to decide whether G can be embedded inside M so that the vertices in every cluster are drawn in the corresponding disc, the edges in every bundle pass only through its corresponding pipe, and every edge crosses the boundary of each disc at most once.Radoslav Fulek, Jan Kynčlwork_ylinuk5p7zf4xejevb4ab2olxqTue, 30 Aug 2022 00:00:00 GMTThe decidability of the genus of regular languages and directed emulators
https://scholar.archive.org/work/p5mexodqgfam7fupxkkuhg2uqe
The article continues our study of the genus of a regular language L, defined as the minimal genus among all genera of all finite deterministic automata recognizing L. Here we define and study two closely related tools on a directed graph: directed emulators and automatic relations. A directed emulator morphism essentially encapsulates at the graph-theoretic level an epimorphism onto the minimal deterministic automaton. An automatic relation is the graph-theoretic version of the Myhill-Nerode relation. We show that an automatic relation determines a directed emulator morphism and respectively, a directed emulator morphism determines an automatic relation up to isomorphism. Consider the set S of all directed emulators of the underlying directed graph of the minimal deterministic automaton for L. We prove that the genus of L is G ∈ Smin g(G). We also consider the more restrictive notion of directed cover and prove that the genus of L is reached in the class of directed covers of the underlying directed graph of the minimal deterministic automaton for L. This stands in sharp contrast to undirected emulators and undirected covers which we also consider. Finally we prove that if the problem of determining the minimal genus of a directed emulator of a directed graph has a solution then the problem of determining the minimal genus of an undirected emulator of an undirected graph has a solution.Guillaume Bonfante, Florian Deloupwork_p5mexodqgfam7fupxkkuhg2uqeSat, 27 Aug 2022 00:00:00 GMTAsymmetrizing cost and density of vertex-transitive cubic graphs
https://scholar.archive.org/work/jadmh2jyzjbdfggptggkoqtcgq
A set S of vertices in a graph G with nontrivial automorphism group is distinguishing if the identity mapping is the only automorphism that preserves S as a set. If such sets exist, then their minimum cardinality is the distinguishing cost ρ(G) of G. A closely related concept is the distinguishing density δ(G). For finite G it is the quotient of ρ(G) by the order of G. We consider connected, vertex-transitive, cubic graphs G and show that either ρ(G) ≤ 5 or ρ(G) = ∞ and δ(G) = 0 if G has one or three arc-orbits, or two arc-orbits and vertex-stabilisers of order at most 2. For the case of two arc-orbits and vertex stabilizers of order > 2 we show the existence of finite graphs with ρ(G) > 5 and infinite graphs with δ(G) > 0. We also prove that two well known results about finite, vertex-transitive, cubic graphs hold without the finiteness condition and construct infinitely many cubic GRRs.Wilfried Imrich, Thomas Lachmann, Thomas W. Tucker, Gundelinde M. Wiegelwork_jadmh2jyzjbdfggptggkoqtcgqWed, 24 Aug 2022 00:00:00 GMTDiscrete group actions on 3-manifolds and embeddable Cayley complexes
https://scholar.archive.org/work/pdv2igggjbh4vo5iwnxzekfpji
We prove that a group Γ admits a discrete topological (equivalently, smooth) action on some simply-connected 3-manifold if and only if Γ has a Cayley complex embeddable – with certain natural restrictions – in one of the following four 3-manifolds: (i) 𝕊^3, (ii) ℝ^3, (iii) 𝕊^2 ×ℝ, (iv) the complement of a tame Cantor set in 𝕊^3.Agelos Georgakopoulos, George Kontogeorgiouwork_pdv2igggjbh4vo5iwnxzekfpjiSun, 21 Aug 2022 00:00:00 GMTMethods of classical and free probability
https://scholar.archive.org/work/xttjetydqzgqzklt4d5nfckbee
This is a joint introduction to classical and free probability, which are twin sisters. We discuss in detail the foundations and main results of both theories, by insisting on their common features, and by using a light formalism, based on standard calculus. We include as well a brief discussion of more advanced aspects.Teo Banicawork_xttjetydqzgqzklt4d5nfckbeeTue, 16 Aug 2022 00:00:00 GMTCharacterizing families of graph manifolds via suitable classes of simple fold maps into the plane and embeddability of the Reeb spaces in some 3-dimensional manifolds
https://scholar.archive.org/work/sz5vyv4sdzel3phpzj5iqc6i6a
Graph manifolds form important classes of 3-dimensional closed and orientable manifolds. For example, Seifert manifolds are graph manifolds where hyperbolic manifolds are not. In applying singularity theory of differentiable maps to understanding global topologies of manifolds, graph manifolds have been shown to be characterized as ones admitting so-called simple fold maps into the plane of explicit classes by Saeki and the author. The present paper presents several related new results. Fold maps are higher dimensional variants of Morse functions and simple ones form simple classes, generalizing the class of general Morse functions. Such maps into the plane on 3-dimensional closed and orientable manifolds induce quotient maps to so-called simple polyhedra with no vertices, which are 2-dimensional. This is also closely related to the theory of shadows of 3-dimensional manifolds. We also discuss invariants for graph manifolds via embeddability of these polyhedra in some 3-dimensional manifolds.Naoki Kitazawawork_sz5vyv4sdzel3phpzj5iqc6i6aMon, 15 Aug 2022 00:00:00 GMTBranched surfaces homeomorphic to Reeb spaces of simple fold maps
https://scholar.archive.org/work/qnjwt5phlfbn7fl3m3wzkdxb4u
Classes of branched surfaces extend the classes of surfaces or 2-dimensional manifolds satisfying suitable properties and defined in various manners. Reeb spaces of smooth maps of suitable classes into surfaces whose codimensions are negative are regarded as branched surfaces. They are the spaces of all connected components of preimages and natural quotient spaces of the manifolds of the domains. They are defined for general smooth maps and important topological objects in differential topology. They also play important roles in applied or applications of mathematics such as projections in data analysis and visualizations. The present paper concerns global topologies of branched surfaces and explicit construction of canonically obtained maps from the branched surfaces into surfaces of the targets via fundamental operations. The class of these induced maps extends the class of smooth immersions of compact surfaces into surfaces with no boundaries. It is also regarded as a variant of the class of so-called generic smooth maps between these surfaces. We study so-called "geography" of such maps as a natural, important and new study and also study global topological properties of the branched surfaces such as embeddability into 3-dimensional closed and connected manifolds.Naoki Kitazawawork_qnjwt5phlfbn7fl3m3wzkdxb4uSat, 13 Aug 2022 00:00:00 GMTPrimitive permutation groups of degree 3p
https://scholar.archive.org/work/medfr4dq4zbrhdutpcnqyhs55e
This paper presents an analysis of primitive permutation groups of degree 3p, where p is a prime number, analogous to H. Wielandt's treatment of groups of degree 2p. It is also intended as an example of the systematic use of combinatorial methods as surveyed in 6 for distilling information about a permutation group from knowledge of the decomposition of its character. The work is organised into three parts. Part I contains the lesser half of the calculation, the determination of the decomposition of the permutation character. Part II contains a survey of the combinatorial methods and, based on these methods, the major part of the calculation. Part III ties up loose ends left earlier in the paper and gives a tabulation of detailed numerical results.Peter M. Neumannwork_medfr4dq4zbrhdutpcnqyhs55eThu, 04 Aug 2022 00:00:00 GMTKilling a Vortex
https://scholar.archive.org/work/r5tvlz2axnb6hfulsufy2huzhm
The Structural Theorem of the Graph Minors series of Robertson and Seymour asserts that, for every t∈ℕ, there exists some constant c_t such that every K_t-minor-free graph admits a tree decomposition whose torsos can be transformed, by the removal of at most c_t vertices, to graphs that can be seen as the union of some graph that is embeddable to some surface of Euler genus at most c_t and "at most c_t vortices of depth c_t". Our main combinatorial result is a "vortex-free" refinement of the above structural theorem as follows: we identify a (parameterized) graph H_t, called shallow vortex grid, and we prove that if in the above structural theorem we replace K_t by H_t, then the resulting decomposition becomes "vortex-free". Up to now, the most general classes of graphs admitting such a result were either bounded Euler genus graphs or the so called single-crossing minor-free graphs. Our result is tight in the sense that, whenever we minor-exclude a graph that is not a minor of some H_t, the appearance of vortices is unavoidable. Using the above decomposition theorem, we design an algorithm that, given an H_t-minor-free graph G, computes the generating function of all perfect matchings of G in polynomial time. This algorithm yields, on H_t-minor-free graphs, polynomial algorithms for computational problems such as the dimer problem, the exact matching problem, and the computation of the permanent. Our results, combined with known complexity results, imply a complete characterization of minor-closed graphs classes where the number of perfect matchings is polynomially computable: They are exactly those graph classes that do not contain every H_t as a minor. This provides a sharp complexity dichotomy for the problem of counting perfect matchings in minor-closed classes.Dimitrios M. Thilikos, Sebastian Wiederrechtwork_r5tvlz2axnb6hfulsufy2huzhmSun, 31 Jul 2022 00:00:00 GMTA web basis of invariant polynomials from noncrossing partitions
https://scholar.archive.org/work/hmdugr5tafchlnlqg6ccvphimi
The irreducible representations of symmetric groups can be realized as certain graded pieces of invariant rings, equivalently as global sections of line bundles on partial flag varieties. There are various ways to choose useful bases of such Specht modules S^λ. Particularly powerful are web bases, which make important connections with cluster algebras and quantum link invariants. Unfortunately, web bases are only known in very special cases – essentially, only the cases λ=(d,d) and λ=(d,d,d). Building on work of B. Rhoades (2017), we construct an apparent web basis of invariant polynomials for the 2-parameter family of Specht modules with λ of the form (d,d,1^ℓ). The planar diagrams that appear are noncrossing set partitions, and we thereby obtain geometric interpretations of earlier enumerative results in combinatorial dynamics.Rebecca Patrias, Oliver Pechenik, Jessica Strikerwork_hmdugr5tafchlnlqg6ccvphimiMon, 18 Jul 2022 00:00:00 GMTOn the generation of some Lie-type geometries
https://scholar.archive.org/work/dvrgrr5xqnhlfbrll4mjqtsm4q
Let X_n(K) be a building of Coxeter type X_n = A_n or X_n = D_n defined over a given division ring K (a field when X_n = D_n). For a non-connected set J of nodes of the diagram X_n, let Γ(K) = Gr_J(X_n(K)) be the J-Grassmannian of X_n(K). We prove that Γ(K) cannot be generated over any proper sub-division ring K_0 of K. As a consequence, the generating rank of Γ(K) is infinite when K is not finitely generated. In particular, if K is the algebraic closure of a finite field of prime order then the generating rank of Gr_1,n(A_n(K)) is infinite, although its embedding rank is either (n+1)^2-1 or (n+1)^2.Ilaria Cardinali, Luca Giuzzi, Antonio Pasiniwork_dvrgrr5xqnhlfbrll4mjqtsm4qThu, 14 Jul 2022 00:00:00 GMTThe complement of enhanced power graph of a finite group
https://scholar.archive.org/work/mjrrqxzmebe3zhlgwv246yy6fa
The enhanced power graph 𝒫_E(G) of a finite group G is the simple undirected graph whose vertex set is G and two distinct vertices x, y are adjacent if x, y ∈⟨ z ⟩ for some z ∈ G. In this article, we give an affirmative answer of the question posed by Cameron [6] which states that: Is it true that the complement of the enhanced power graph 𝒫̅_̅E̅(̅G̅)̅ of a non-cyclic group G has only one connected component apart from isolated vertices? We classify all finite groups G such that the graph 𝒫̅_̅E̅(̅G̅)̅ is bipartite. We show that the graph 𝒫̅_̅E̅(̅G̅)̅ is weakly perfect. Further, we study the subgraph 𝒫̅_̅E̅(̅G̅^̅*̅)̅ of 𝒫̅_̅E̅(̅G̅)̅ induced by all the non-isolated vertices of 𝒫̅_̅E̅(̅G̅)̅. We classify all finite groups G such that the graph is 𝒫̅_̅E̅(̅G̅^̅*̅)̅ is unicyclic and pentacyclic. We prove the non-existence of finite groups G such that the graph 𝒫̅_̅E̅(̅G̅^̅*̅)̅ is bicyclic, tricyclic or tetracyclic. Finally, we characterize all finite groups G such that the graph 𝒫̅_̅E̅(̅G̅^̅*̅)̅ is outerplanar, planar, projective-planar and toroidal, respectively.Parveen, Jitender Kumarwork_mjrrqxzmebe3zhlgwv246yy6faMon, 11 Jul 2022 00:00:00 GMTRandom Colorings in Manifolds
https://scholar.archive.org/work/rwyyk2xcojb4ph7j5j5aklvr3m
We develop a general method for constructing random manifolds and submanifolds in arbitrary dimensions. The method is based on associating colors to the vertices of a triangulated manifold, as in recent work for curves in 3-dimensional space by Sheffield and Yadin (2014). We determine conditions on which submanifolds can arise, in terms of Stiefel-Whitney classes and other properties. We then consider the random submanifolds that arise from randomly coloring the vertices. Since this model generates submanifolds, it allows for studying properties and using tools that are not available in processes that produce general random subcomplexes. The case of 3 colors in a triangulated 3-ball gives rise to random knots and links. In this setting, we answer a question raised by de Crouy-Chanel and Simon (2019), showing that the probability of generating an unknot decays exponentially. In the general case of k colors in d-dimensional manifolds, we investigate the random submanifolds of different codimensions, as the number of vertices in the triangulation grows. We compute the expected Euler characteristic, and discuss relations to homological percolation and other topological properties. Finally, we explore a method to search for solutions to topological problems by generating random submanifolds. We describe computer experiments that search for a low-genus surface in the 4-dimensional ball whose boundary is a given knot in the 3-dimensional sphere.Chaim Even-Zohar, Joel Hasswork_rwyyk2xcojb4ph7j5j5aklvr3mFri, 08 Jul 2022 00:00:00 GMTA practical algorithm for the computation of the genus
https://scholar.archive.org/work/yxrc52qinbck7gfbowzt5qy64i
We describe a practical algorithm to compute the (orientable) genus of a graph, give results of the program implementing this algorithm, and compare the performance to existing algorithms. The aim of this algorithm is to be fast enough for many applications instead of focusing on the theoretical asymptotic complexity. Apart from the specific problem and the results, the article can also be seen as an example how some design principles used to carefully develop and implement standard backtracking algorithms can still result in very competitive programs.Gunnar Brinkmannwork_yxrc52qinbck7gfbowzt5qy64iFri, 01 Jul 2022 00:00:00 GMTA density bound for triangle-free 4-critical graphs
https://scholar.archive.org/work/xdujgwhw45csvey3lmxae2l6gu
We prove that every triangle-free 4-critical graph G satisfies e(G) ≥5v(G)+2/3. This result gives a unified proof that triangle-free planar graphs are 3-colourable, and that graphs of girth at least five which embed in either the projective plane, torus, or Klein Bottle are 3-colourable, which are results of Grötzsch, Thomassen, and Thomas and Walls. Our result is nearly best possible, as Davies has constructed triangle-free 4-critical graphs G such that e(G) = 5v(G) + 4/3. To prove this result, we prove a more general result characterizing sparse 4-critical graphs with few vertex-disjoint triangles.Benjamin Moore, Evelyne Smith-Robergework_xdujgwhw45csvey3lmxae2l6guThu, 30 Jun 2022 00:00:00 GMTUnsolved Problems in Group Theory. The Kourovka Notebook
https://scholar.archive.org/work/fhii5oyzvrb7vpeun6rsfapu2i
This is a collection of open problems in group theory proposed by hundreds of mathematicians from all over the world. It has been published every 2-4 years in Novosibirsk since 1965. This is the 20th edition, which contains 126 new problems and a number of comments on problems from the previous editions.E. I. Khukhro, V. D. Mazurovwork_fhii5oyzvrb7vpeun6rsfapu2iMon, 27 Jun 2022 00:00:00 GMT