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Oleg Viro

Publications and source records attributed to Oleg Viro.

At least 19 recordsLinked to original sources
arXiv2025

Complex slices on a real variety

Let $X$ be a real algebraic variety with set of complex points $X_{\mathbb C}$ and set of real points $X_{\mathbb R}$. A complex slice of $X$ is a transverse intersection of $X_{\mathbb R}$ with a complex subvariety $V$ of $X_{\mathbb C}$. Complex slices are real algebraic varieties of a very special kind. They are cooriented, realize an integer cohomology class. A codimension 2 projective variety is a slice, iff it is a base of pencil of real algebraic hypersurfaces. We prove an upper bound for the linking number of a real projective curve bounding in its complexification with a slice of codimension two.

math.AG↗
arXiv2020

Rokhlin's signature theorems

This note is written for a book dedicated to outstanding St-Petersburg mathematicians and timed to the ICM-2022 in St-Petersburg. In accordance with the plan of ICM-organizers, we try to tell about one of the most prominent Rokhlin's achievements in an accessible form and respecting the allowed volume.

math.HO↗
arXiv2020

Fundamental group in the projective knot theory

In this paper, properties of a link $L$ in the projective space $\mathbb R P^3$ are related to properties of its group $π_1(\mathbb R P^3\smallsetminus L)$: $L$ is isotopic to a projective line if and only if $π_1(\mathbb R P^3\smallsetminus L)=\mathbb Z$. $L$ is isotopic to an affine circle if and only if $π_1(\mathbb R P^3\smallsetminus L)=\mathbb Z*\mathbb Z_{/2}$. $L$ is isotopic to a link disjoint from a projective plane if and only if $π_1(\mathbb R P^3\smallsetminus L)$ contains a non-trivial element of order two. A simple algorithm which finds a system of generators and relations for $π_1(\mathbb R P^3\smallsetminus L)$ in terms of a link diagram of $L$ is provided.

math.GT↗
arXiv2019

Characterization of affine links in the projective space

A projective link is a smooth closed 1-submanifold of the real projective space of dimension three. A projective link is said to be affine if it is isotopic to a link, which does not intersect some projective plane. The main result: a projective link is affine if and only if the fundamental group of its complement contains a non-trivial element of order two.

math.GT↗
arXiv2015

Biflippers and head to tail composition rules

A new graphical calculus for operating with isometries of low dimensional spaces of classical geometries is proposed. It is similar to a well-known graphical representation for vectors and translations in an affine space. Instead of arrows, we use biflippers, which are arrows framed at the end points with subspaces. The head to tail addition of vectors and composition of translations is generalized to head to tail composition rules for isometries.

math.MG↗
arXiv2014

Defining relations for reflections. I

An idea to present a classical Lie group of positive dimension by generators and relations sounds dubious, but happens to be fruitful. The isometry groups of classical geometries admit elegant and useful presentations by generators and relations closely related to geometry. They allow to make fast and efficient geometric calculations. In this paper simple presentations of the isometry groups of Euclidean plane, 2-sphere, the real projective plane and groups SO(3), O(n) are introduced.

math.MG↗
arXiv2010

Hyperfields for Tropical Geometry I. Hyperfields and dequantization

New hyperfields, that is fields in which addition is multivalued, are introduced and studied. In a separate paper these hyperfields are shown to provide a base for the tropical geometry. The main hyperfields considered here are classical number sets, such as the set of complex numbers, the set of real numbers, and the set of real non-negative numbers, with the usual multiplications, but new, multivalued additions. The new hyperfields are related with the classical fields and each other by dequantisations. For example, the new complex tropical field is a dequantization of the field of complex numbers.

math.AG↗
arXiv2010

Twisted acyclicity of a circle and link signatures

Homology of the circle with non-trivial local coefficients is trivial. From this well-known fact we deduce geometric corollaries concerning links of codimension two. In particular, the Murasugi-Tristram signatures are extended to invariants of links formed of arbitrary oriented closed codimension two submanifolds of an odd-dimensional sphere. The novelty is that the submanifolds are not assumed to be disjoint, but are transversal to each other, and the signatures are parametrized by points of the whole torus. Murasugi-Tristram inequalities and their generalizations are also extended to this setup.

math.GT↗
arXiv2006

Orientations of Chord Diagrams and Khovanov Homology

By adding or removing appropriate structures to Gauss diagram, one can create useful objects related to virtual links. In this paper few objects of this kind are studied: twisted virtual links generalizing virtual links; signed chord diagrams staying halfway between twisted virtual links and Kauffman bracket / Khovanov homology; alternatable virtual links intermediate between virtual and classical links. The most profound role here belongs to a structure that we dare to call orientation of chord diagram. Khovanov homology is generalized to oriented signed chord diagrams and links in oriented thickened surface such that the link projection realizes the first Stiefel-Whitney class of the surface. After this paper was published, V.O.Manturov succeeded in extending Khovanov homology with arbitrary coefficients to arbitrary virtual links, see arXiv: math.GT/0601152.

math.GT↗
arXiv2006

Configurations of skew lines

This paper is an updated version of a survey on projective configurations of subspaces in general position. The preceding version was published in Russian in 1989 and in English in 1990 (in Leningrad Math. J.) opening a new section ``Light reading for the professional''. The paper is written in the form of introduction to the subject, with much of the material accessible to advanced high school students. However, in the part of the survey concerning configurations of lines in general position in the three-dimensional space the exposition is free from any background restrictions. We have added few new results, fixed few misprints and terminological inaccuracies and expanded the reference list. Notice that some of the results presented in the paper appeared in other papers without appropriate references.

math.GT↗
arXiv2006

Patchworking real algebraic varieties

Patchworking is a construction of a one-parameter family of real algebraic hypersurfaces. For sufficiently small positive values of the parameter, the hypersurfaces can be obtained by gluing of given hypersurfaces topologically. The author invented patchworking in 1979-81 and used it for constructing of real plane algebraic curves with complicated prescribed topology. In particular, it helped to complete isotopy classification of nonsingular plane projective real algebraic curves of degree 7. A special case of the patchworking, combinatorial patchworking, can be considered as Litvinov-Maslov quantization of a tropical variety. Due to its simplicity, combinatorial patchworking is better known than the general one. This paper is the original presentation of the patchworking, in its full generality.

math.AG↗
arXiv2006

Complex orientations of real algebraic surfaces

We study natural additional structures on real algebraic surfaces with trivial first homology mod 2 of the complexification. If the set of real points realizes the zero of the second homology mod 2 of the complexification, then the set of real points is equipped with a pair of opposite orientations and a Spin structure. If the set of real points realizes the same homology class as the complexification of a real curve on the surface, then the complement of the curve in set of real points is equipped a pair of opposite orientations, which do not extend across the curve, and the whole set of real points is equipped with a Pin^- structure. These constructions are similar to the complex orientations of real algebraic curves dividing their complexifications and generalize to high dimensions.

math.AG↗
arXiv2002

Quantum Relatives of Alexander Polynomial

The multivariable Conway function is generalized to oriented framed trivalent graphs equipped with additional structure (coloring). This is done via refinements of Reshetikhin-Turaev functors based on irreducible representations of quantized gl(1|1) and sl(2). The corresponding face state sum models for the generalized Conway function are presented.

math.GT↗
arXiv2002

Remarks on definition of Khovanov homology

Mikhail Khovanov in math.QA/9908171 defined, for a diagram of an oriented classical link, a collection of groups numerated by pairs of integers. These groups were constructed as homology groups of certain chain complexes. The Euler characteristics of these complexes are coefficients of the Jones polynomial of the link. The goal of this note is to rewrite this construction in terms more friendly to topologists. A version of Khovanov homology for framed links is introduced. For framed links whose Kauffman brackets are involved in a skein relation, these homology groups are related by an exact sequence.

math.GT↗
arXiv2000

Dequantization of real algebraic geometry on logarithmic paper

On logarithmic paper some real algebraic curves look like smoothed broken lines. Moreover, the broken lines can be obtained as limits of those curves. The corresponding deformation can be viewed as a quantization, in which the broken line is a classical object and the curves are quantum. This generalizes to a new connection between algebraic geometry and the geometry of polyhedra, which is more straight-forward than the other known connections and gives a new insight into constructions used in the topology of real algebraic varieties.

math.AG↗
arXiv2000

Encomplexing the writhe

A detailed version of preprint "Self-linking number of a real algebraic link" by the same author, alg-geom/9410030. For a nonsingular real algebraic curve in 3-dimensional projective space or 3-sphere, a new integer-valued characteristic is introduced. It is invariant under rigid isotopy and multiplied by -1 under mirror reflections. In a sense, it is a Vassiliev invariant of degree 1 and a counterpart of a link diagram writhe. For a regular complete intersection this invariant vanishes, while for rational knots of degree d it takes all the values between -(d-1)(d-2)/2 and (d-1)(d-2)/2.

math.AG↗