arXiv subjects
Publications and source records attributed to Vassily Olegovich Manturov.
We construct a novel invariant of braids and knots, called the secant-quandle (SQ), derived from homotopy classes of generic secants and generic horizontal trisecants. This invariant provides a natural generalization of the usual knot quandle, capturing richer topological information.
We introduce a broad family of groups $G(Γ)$ associated with transverse stratifications of moduli spaces. For a wide class of stratifications satisfying a natural local condition (transverse niceness), we construct a homomorphism from the fundamental group of the moduli space to the corresponding group $G(Γ)$. This general theorem, which is proved in the companion paper \cite{Manturov2026}, provides a far-reaching generalization of the $G_n^k$-theory. The main goal of the present paper is to discuss numerous examples and situations where the transverse niceness property actually occurs. We describe various moduli spaces and stratifications that are transversely nice. In particular, we study the hierarchy of configuration spaces of points in the plane defined by algebraic curves and establish the transversality property for conic configurations. The general hierarchy problem is formulated in the final section.
In this work, we construct a generalization of the $G_n^k$-theory to the case of an arbitrary hypergraph. The case of spherical braids is considered separately, using the stratification of the moduli space $\mathcal{M}_n(S^2)$ and the hypergraph $Γ_n^{\mathrm{sph}}$ encoding projective constraints. In contrast to the original $G_{n}^{k}$ theory, where codimension-one properties are determined by exactly $k$ particles, the present work considers various cases corresponding to strata of codimension~$1$. These groups admit nice maps to free products of cyclic groups. Among unsolved problems, we emphasize the question how the above construction works for abelian varieties and, in particular, for elliptic curves.
In the present paper, we discuss a way of generalising Vassiliev knot invariants and weight systems to framed chord diagrams having framing 0 and 1.
We consider a certain modification of the group $G^3_n$ which describes dynamics of point configurations, in particular braids, and define a representation of the modified $G^3_n$. The braid representation induced is powerful enough to detect the kernel of the Burau representation.
We present a way of using shear coordinates in hyperbolic geometry to get invariants of braids. This method also has a tropical analogue.
In the present paper we discuss four ways of looking at rhombile tilings: stacking 3-dimensional cubes, elements of groups, and configurations of lines and points.
In the present paper, we construct an invariant of braids in the real projective plane which corresponds to an ``action'' of braids on certain graphs in $\R{}P^{2}$ with labels. This paper is a sequel of papers \cite{M},\cite{KM}. It demonstrates once more that solutions to the octagon relation in various forms give rise to invariants of braids.
In \cite{ManturovNikonovMay2023,ManturovWanMay2023} the author discovered a very general principle (called {\em the photography principle}) which allows one: a) To solve various equations (like pentagon equation) b) To construct invariants of manifolds. The advantage of that principle is that it deals with a very general notion of {\em data} and {\em data transmission} which may be of any kind. In the present paper, we show that the definition of the Dijkgraaf-Witten invariants of manifolds can be thought of as an evidence of the above principle.
In the present paper, we consider two applications of the pentagon equation. The first deals with actions of flips on edges of triangulations labelled by rational functions in some variables. The second can be formulated as a system of linear equations with variables corresponding to triangles of a triangulation. The general method says that if there is some general {\em data} (say, edge lengths or areas) associated with {\em states} (say, triangulations) and a general {\em data transformation rule} (say, how lengths or areas are changed under flips) then after returning to the initial state we recover the initial data.
Similar pictures appear in various branches of mathematics. Sometimes this similarity gives rise to deep theorems. Mentioning such a similarity between hexagonal tilings, cubes in 3-space, configurations of lines and braid groups, we prove that braids act on configurations of lines.
We introduce a family of groups $Γ_n^k$ for integer parameters $n>k$. These groups originate from discussion of braid groups on $2$-surfaces. On the other hand, they turn out to be related to 3-manifolds (in particular, they lead to new relationships between braids and manifolds), triangulations (ideal triangulations) cluster algebras, dynamics of moving points, quivers, hyperbolic structures, tropical geometry, and, probably, many other areas still to be discovered. Among crucial reason of this importance of groups $Γ_{n}^{4}$ we mention the Ptolemy relation, Pentagon relation, cluster algebra, Stasheff polytope.
The aim of this article is to prove that the kernel of the map from the pure braid group $PB_{n},n\ge 4$ to the group $G_{n}^{3}$ consists of full twist braids and their exponents. The proof consists of two parts. The first part which deals with $n=4$ relies on the crucial tool in this construction having its own interest is the {\em realisability projection} saying that if two {\em realisable} $G_{4}^{3}$-elements are equivalent then they are equivalent by a sequence of realisable ones. The second part (an easy one) uses induction on $n$.
The aim of the present note is to enhance groups $G_{n}^{3}$ and to construct new invariants of classical braids. In particular, we construct invariants valued in $G_{N}^{2}$ groups. In groups $G_{n}^{2}$, the identity problem is solved, besides, their structure is much simpler than that of $G_{n}^{3}$. I am grateful to Huyue Yan for pointing out a small mistake in the previous version.
Knot concordance plays a crucial role in the low dimensional topology. We propose a very elementary techniques which allows one to construct a lot of sliceness obstructions for knots in the full torus. Our approach deals with group theoretical techniques; it is completely combinatorial, and the groups are very easy to deal with.
We construct an invariant of virtual knots which is a sliceness obstruction and sensitive to the $Δ$-move. This invariants works if $\Z_{2}\oplus \Z_{2}$-index of chords is present.
The aim of this paper is to realise the techniques of picture-valued invariants and invariants valued in free groups for long knots in the full torus. Such knots and links are of a particular interest because of their relation to Legendrian knots, knotoids, 3-manifolds and many other objects. Invariants constructed in the paper are powerful and easy to compare. This paper is a sequel of [6]. Long knots naturally appear in the study of classical knots [1, 8].
In the present paper, we study several complex manifolds by using the following idea. First, we construct a certain moduli space and study the fundamental group of this space. This fundamental group is naturally mapped to the groups $G_{n}^{k}$ and $Γ_{n}^{k}$. This is the step towards "complexification" of the $G_{n}^{k}$ and $Γ_{n}^{k}$ approach first developed in \cite{2019arXiv190508049M}.