arXiv subjects

Oleg Karpenkov

Publications and source records attributed to Oleg Karpenkov.

At least 19 recordsLinked to original sources

arXiv2026

Wug-snake graphs and Markov numbers of matrix semigroups

Classically, Markov numbers are recovered as perfect matching numbers of domino snake graphs. We extend this correspondence by introducing weighted universal generalised snake graphs, or wug-snake graphs. These are weighted ordered bipartite graphs whose perfect matching sequences encode linear recurrences. To every wug-snake graph we associate a continuant matrix and prove that the determinant of this matrix equals the weighted perfect matching sum. We then introduce polyomino wug-tiles, bodies of wug-snake graphs that act linearly on state vectors. Every integer matrix admits a canonical polyomino wug-tile. Our main result identifies the wug-snake determinant of a tile representing a matrix $A$ with the Markov-Davenport form of $A$. Consequently, algebraic and geometric Markov numbers of matrices and matrix semigroups can be expressed as weighted perfect matching determinants. Further we define Frobenius maps for matrix semigroups and discuss examples recovering classical Markov numbers and higher-dimensional lattice realisations.

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arXiv2026

Classifying integer tilings and hypertilings

There are two objectives to this work: to classify all tame integer tilings and to classify all tame integer hypertilings. Motivation for the first objective comes from Conway and Coxeter's modelling of positive integer friezes using triangulated polygons, which has received significant attention since the discovery of cluster algebras by Fomin and Zelevinsky in 2002. Assem, Reutenauer, and Smith introduced $\text{SL}_2$-tilings as generalisations of friezes, and Bessenrodt, Holm, and Jørgensen classified positive integer $\text{SL}_2$-tilings using infinite triangulated polygons. Here we consider $N$-tilings, of which $\text{SL}_2$-tilings are the case $N=1$. We provide a geometric model for all tame integer $N$-tilings using a generalisation of the Farey graph in the hyperbolic plane. Highlights of this model include classifications of all positive integer $N$-tilings and of all positive rational friezes, with entries encoded by lambda lengths or weight data of triangulated polygons. The second objective is motivated by Bhargava's celebrated study of binary quadratic forms using integer cubes and by an observation of Demonet et al. that there is essentially only one three-dimensional positive integer tiling with $\text{SL}_2$ cross sections. We consider a richer class of three-dimensional tilings, which we call hypertilings, using the Cayley hyperdeterminant. We classify all tame integer hypertilings using generalised Farey graphs; remarkably, those with Cayley hyperdeterminant 1 prove to have a simple description in terms of triple Hadamard products of integer pairs.

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arXiv2025

Geometry of multidimensional Farey summation algorithm and frieze patterns

In this paper we develop a new geometric approach to subtractive continued fraction algorithms in high dimensions. We adapt a version of Farey summation to the geometric techniques proposed by F. Klein in 1895. More specifically we introduce Farey polyhedra and their sails that generalise respectively Klein polyhedra and their sails, and show similar duality properties of the Farey sail integer invariants. The construction of Farey sails is based on the multidimensional generalisation of the Farey tessellation provided by a modification of the continued fraction algorithm introduced by R. W. J. Meester. We classify Farey polyhedra in the combinatorial terms of prismatic diagrams. Prismatic diagrams extend boat polygons introduced by S. Morier-Genoud and V. Ovsienko in the two-dimensional case. As one of the applications of the new theory we get a multidimensional version of Conway-Coxeter frieze patterns. We show that multidimensional frieze patterns satisfy generalised Ptolemy relations.

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arXiv2025

On Euclidean Algorithms for oriented linear Grassmanians

In this paper we study Euclidean algorithms and the corresponding continued fractions for oriented linear Grassmanians $G(k,n)$. We propose two algorithms: Maximal Element Elimination algorithm and Minimal Element Elimination algorithm. The first algorithm reduces the absolute maximal value of the Plücker coordinates; the algorithm works only in $G(2,n)$. The second algorithm eliminates the Plücker coordinate with the smallest absolute values, while all other coordinates may increase; the algorithm works for arbitrary $G(2,n)$. We discuss basic features of these algorithms and formulate several natural open questions for further studies.

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arXiv2025

Stressability of Semi-Discrete Frameworks

In this paper we study the stressability of semi-discrete frameworks in the plane which are generated by a discrete sequence of smooth curves. We characterize their stressability property by the existence of stresses fulfilling certain difference-differential equation. In particular, we define a semi-discrete height function which we use to generate liftings. Furthermore, we show a semi-discrete analogue of the Maxwell-Cremona lifting property which implies that the stressable semi-discrete frameworks in the plane are precisely the orthogonal projections of semi-discrete conjugate surfaces in 3-space. Finally, we discuss geometric implications for frameworks with vanishing boundary forces and characterize the liftability of frameworks which only consist of two neighboring curves forming one strip.

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arXiv2025

Liftings of surfaces in the plane

In this note we provide a topological definition of Maxwell-Cremona liftings for non-planar frameworks of surfaces (both oriented and non-oriented). In the non-oriented case we give an estimate on the dimension of self-stresses, when the frameworks will posses a non-trivial lifting.

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arXiv2024

Circumscribed Circles in Integer Geometry

Integer geometry on a plane deals with objects whose vertices are points in $\mathbb Z^2$. The congruence relation is provided by all affine transformations preserving the lattice $\mathbb Z^2$. In this paper we study circumscribed circles in integer geometry. We introduce the notions of integer and rational circumscribed circles of integer sets. We determine the conditions for a finite integer set to admit an integer circumscribed circle and describe the spectra of radii for integer and rational circumscribed circles.

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arXiv2024

Klein-Arnold tensegrities

In this paper, we introduce new classes of infinite and combinatorially periodic tensegrities, derived from algebraic multidimensional continued fractions in the sense of F. Klein. We describe the stress coefficients on edges through integer invariants of these continued fractions, as initiated by V.I. Arnold, thereby creating a novel connection between geometric rigidity theory and the geometry of continued fractions. Remarkably, the new classes of tensegrities possess rational self-stress coefficients. To establish the self-stressability of the frameworks, we present a projective version of the classical Maxwell-Cremona lifting principle, a result of independent interest.

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arXiv2024

Farey Bryophylla

The construction of the Farey tessellation in the hyperbolic plane starts with a finitely generated group of symmetries of an ideal triangle, i.e. a triangle with all vertices on the boundary. It induces a remarkable fractal structure on the boundary of the hyperbolic plane, encoding every element by the continued fraction related to the structure of the tessellation. The problem of finding a generalisation of this construction to the higher dimensional hyperbolic spaces has remained open for many years. In this paper we make the first steps towards a generalisation in the three-dimensional case. We introduce conformal bryophylla, a class of subsets of the boundary of the hyperbolic 3-space which possess fractal properties similar to the Farey tessellation. We classify all conformal bryophylla and study the properties of their limiting sets.

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arXiv2023

A differential approach to Maxwell-Cremona liftings

In 1864, J. C. Maxwell introduced a link between self-stressed frameworks in the plane and piecewise linear liftings to 3-space. This connection has found numerous applications in areas such as discrete geometry, control theory and structural engineering. While there are some generalisations of this theory to liftings of $d$-complexes in $d$-space, extensions for liftings of frameworks in $d$-space for $d\geq 3$ have been missing. In this paper, we introduce and study differential liftings on general graphs using differential forms associated with the elements of the homotopy groups of the complements to the frameworks. Such liftings play the role of integrands for the classical notion of liftings for planar frameworks. We show that these differential liftings have a natural extension to self-stressed frameworks in higher dimensions. As a result we generalise the notion of classical liftings to both graphs and multidimensional $k$-complexes in $d$-space ($k=2,\ldots, d$). Finally we discuss a natural representation of generalised liftings as real-valued functions on Grassmannians.

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arXiv2023

Lattice angles of lattice polygons

This paper is dedicated to a lattice analog to the classical ``sum of interior angles of a polygon theorem''. In 2008, the first formula expressing conditions on the geometric continued fractions for lattice angles of triangles was derived, while the cases of $n$-gons for $n > 3$ remained unresolved. In this paper, we provide the complete solution for all integer $n$. The main results are based on recent advances in geometry of continued fractions.

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arXiv2023

$3$D Farey graph, lambda lengths and $SL_2$-tilings

We explore a three-dimensional counterpart of the Farey tessellation and its relations to Penner's lambda lengths and $SL_2$-tilings. In particular, we prove a three-dimensional version of Ptolemy relation, and generalise results of Ian Short to classify tame $SL_2$-tilings over Eisenstein integers in terms of pairs of paths in the 3D Farey graph.

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arXiv2023

Multidimensional integer trigonometry

This paper is dedicated to providing an introduction into multidimensional integer trigonometry. We start with an exposition of integer trigonometry in two dimensions, which was introduced in 2008, and use this to generalise these integer trigonometric functions to arbitrary dimension. We then move on to study the basic properties of integer trigonometric functions. We find integer trigonometric relations for transpose and adjacent simplicial cones, and for the cones which generate the same simplices. Additionally, we discuss the relationship between integer trigonometry, the Euclidean algorithm, and continued fractions. Finally, we use adjacent and transpose cones to introduce a notion of best approximations of simplicial cones. In two dimensions, this notion of best approximation coincides with the classical notion of the best approximations of real numbers.

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arXiv2021

Continued Fraction approach to Gauss Reduction Theory

Jordan Normal Forms serve as excellent representatives of conjugacy classes of matrices over closed fields. Once we knows normal forms, we can compute functions of matrices, their main invariant, etc. The situation is much more complicated if we search for normal forms for conjugacy classes over fields that are not closed and especially for rings. In this paper we study PGL(2,Z)-conjugacy classes of GL(2,Z) matrices. For the ring of integers Jordan approach has various limitations and in fact it is not effective. The normal forms of conjugacy classes of GL(2,Z) matrices are provided by alternative theory, which is known as Gauss Reduction Theory. We introduce a new techniques to compute reduced forms In Gauss Reduction Theory in terms of the elements of certain continued fractions. Current approach is based on recent progress in geometry of numbers. The proposed technique provides an explicit computation of periods of continued fractions for the slopes of eigenvectors.

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arXiv2021

On a periodic Jacobi-Perron type algorithm

In this paper we introduce a new modification of the Jacobi-Perron algorithm in three dimensional case and prove its periodicity for the case of totally-real conjugate cubic vectors. This provides an answer in the totally-real case to the question son algebraic periodicity for cubic irrationalities posed in 1849 by Ch.~Hermite.

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arXiv2021

Geometric criteria for realizability of tensegrities in higher dimensions

In this paper we study a classical Maxwell question on the existence of self-stresses for frameworks, which are called tensegrities. We give a complete answer on geometric conditions of at most $(d+1)$-valent tensegrities in $\mathbb{R}^d$ both in terms of discrete multiplicative 1-forms and in terms of "meet" and "join" operations in the Grassmann-Cayley algebra.

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arXiv2021

On Hermite's problem, Jacobi-Perron type algorithms, and Dirichlet groups

In 1848 Ch.~Hermite asked if there exists some way to write cubic irrationalities periodically. A little later in order to approach the problem C.G.J.~Jacobi and O.~Perron generalized the classical continued fraction algorithm to the three-dimensional case, this algorithm is called now the Jacobi-Perron algorithm. This algorithm is known to provide periodicity only for some cubic irrationalities. In this paper we introduce two new algorithms in the spirit of Jacobi-Perron algorithm: the heuristic algebraic periodicity detecting algorithm and the $\sin^2$-algorithm. The heuristic algebraic periodicity detecting algorithm is a very fast and efficient algorithm, its output is periodic for numerous examples of cubic irrationalities, however its periodicity for cubic irrationalities is not proven. The $\sin^2$-algorithm is limited to the totally-real cubic case (all the roots of cubic polynomials are real numbers). In the recent paper~\cite{Karpenkov2021} we proved the periodicity of the $\sin^2$-algorithm for all cubic totally-real irrationalities. To our best knowledge this is the first Jacobi-Perron type algorithm for which the cubic periodicity is proven. The $\sin^2$-algorithm provides the answer to Hermite's problem for the totally real case (let us mention that the case of cubic algebraic numbers with complex conjugate roots remains open). We conclude this paper with one important application of Jacobi-Perron type algorithms to computation of independent elements in the maximal groups of commuting matrices of algebraic irrationalities.

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arXiv2020

Equilibrium stressability of multidimensional frameworks

We prove an equilibrium stressability criterium for trivalent multidimensional tensegrities. The criterium appears in different languages: (1) in terms of stress monodromies, (2) in terms of surgeries, (3) in terms of exact discrete 1-forms, and (4) in Cayley algebra terms.

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