arXiv subjects
Publications and source records attributed to Vassily Olegovich Manturov.
In the present paper, we proceed the study of framed $4$-graph minor theory initiated in ``Framed $4$-valent Graph Minor Theory I. Intoduction. Planarity Criterion '' and justify the planarity theorem for arbitrary framed 4-graphs; besides, we prove analogous results for embeddability in the projective plane.
The present paper is the first one in the sequence of papers about a simple class of {\em framed $4$-graphs}; the goal of the present paper is to collect some well-known results on planarity and to reformulate them in the language of {\em minors}. The goal of the whole sequence is to prove analogues of the Robertson-Seymour-Thomas theorems for framed $4$-graphs: namely, we shall prove that many minor-closed properties are classified by finitely many excluded graphs. From many points of view, framed $4$-graphs are easier to consider than general graphs; on the other hand, framed $4$-graphs are closely related to many problems in graph theory.
This paper considers *-graphs in which all vertices have degree 4 or 6, and studies the question of calculating the genus of nonorientable surfaces into which such graphs may be embedded. In a previous paper by the authors, the problem of calculating whether a given *-graph in which all vertices have degree 4 or 6 admits a Z2-homologically trivial embedding into a given orientable surface was shown to be equivalent to a problem on matrices. Here we extend those results to nonorientable surfaces. The embeddability condition that we obtain yields quadratic-time algorithms to determine whether a *-graph with all vertices of degree 4 or 6 admits a Z2-homologically trivial embedding into the projective plane or into the Klein bottle.
In the present paper, we construct the generalized Kuperberg bracket for two-component links with one component fibred. We consider a new geometrical complexity for such links and establish minimality of diagrams in a strong sense.
In the present paper, we have found new upper bounds for chromatic numbers for integer lattices and some rational spaces and other lattices. In particular, we have proved that for any concrete critical distance $d$ the chromatic number of $\Z^{n}$ with critical distance $\sqrt{2d}$ has a polynomial growth in $n$ with exponent less than or equal to $d$ (sometimes this estimate is sharp). The same statement is true not only in the Euclidean norm, but also in any $l_{p}$ norm. Besides, we have given concrete estimates for some small dimensions as well as upper bounds for the chromatic number of $\Q_{p}^{n}$, where by $\Q_{p}$ we mean the ring of all rational numbers having denominators not divisible by some prime numbers.
We construct various functorial maps (projections) from virtual knots to classical knots. These maps are defined on diagrams of virtual knots; in terms of Gauss diagram each of them can be represented as a deletion of some chords. The construction relies upon the notion of parity. As corollaries, we prove that the minimal classical crossing number for classical knots.
The present paper produces examples of Gauss diagram formulae for virtual knot invariants which have no analogue in the classical knot case. These combinatorial formulae contain additional information about how a subdiagram is embedded in a virtual knot diagram. The additional information comes from the second author's recently discovered notion of parity. For a parity of flat virtual knots, the new combinatorial formulae are Kauffman finite-type invariants. However, many of the combinatorial formulae possess exotic properties. It is shown that there exists an integer valued virtualization invariant combinatorial formula of order n for every n (i.e. it is stable under the map which changes the direction of one arrow but preserves the sign). Hence, it is not of Goussarov-Polyak-Viro finite-type. Moreover, every homogeneous Polyak-Viro combinatorial formula admits a decomposition into an "even" part and an "odd" part. For the Gaussian parity, neither part of the formula is of GPV finite-type when it is nonconstant on the set of classical knots. In addition, eleven new non-trivial combinatorial formulae of order 2 are presented which are not of GPV finite-type.
The notion of free link is a generalized notion of virtual link. In the present paper we define the group of free braids, prove the Alexander theorem that all free links can be obtained as closures of free braids and prove a Markov theorem, which gives necessary and sufficient conditions for two free braids to have the same free link closure. Our result is expected to be useful in study the topology invariants for free knots and links.
The aim of the present paper is to prove that the minimal number of virtual crossings for some families of virtual knots grows quadratically with respect to the minimal number of classical crossings. All previously known estimates for virtual crossing number were principally no more than linear in the number of classical crossings (or, what is the same, in the number of edges of a virtual knot diagram) and no virtual knot was found with virtual crossing number greater than the classical crossing number.
In \cite {FrKn,Sbornik} it was shown that in some knot theories the crucial role is played by {\em parity}, i.e.\ a function on crossings valued in $\{0,1\}$ and behaving nicely with respect to Reidemeister moves. Any parity allows one to construct functorial mappings from knots to knots, to refine many invariants and to prove minimality theorems for knots. In the present paper, we generalise the notion of parity and construct parities with coefficients from an abelian group rather than $\mathbb{Z}_2$ and investigate them for different knot theories. For some knot theories we show that there is the universal parity, i.e.\ such a parity that any other parity factors through it. We realise that in the case of flat knots all parities originate from homology groups of underlying surfaces and, at the same time, allow one to "localise" the global homological information about the ambient space at crossings. We prove that there is only one non-trivial parity for free knots, the Gaussian parity. At the end of the paper we analyse the behaviour of some invariants constructed for some modifications of parities.
In the present paper, we construct a simple invariant which provides a sliceness obstruction for {\em free knots}. This obstruction provides a new point of view to the problem of studying cobordisms of curves immersed in 2-surfaces, a problem previously studied by Carter, Turaev, Orr, and others. The obstruction to sliceness is constructed by using the notion of {\em parity} recently introduced by the author into the study of virtual knots and their modifications. This invariant turns out to be an obstruction for cobordisms of higher genera with some additional constraints.
Two categorifications are given for the arrow polynomial, an extension of the Kauffman bracket polynomial for virtual knots. The arrow polynomial extends the bracket polynomial to infinitely many variables, each variable corresponding to an integer {\it arrow number} calculated from each loop in an oriented state summation for the bracket. The categorifications are based on new gradings associated with these arrow numbers, and give homology theories associated with oriented virtual knots and links via extra structure on the Khovanov chain complex. Applications are given to the estimation of virtual crossing number and surface genus of virtual knots and links. Key Words: Jones polynomial, bracket polynomial, extended bracket polynomial, arrow polynomial, Miyazawa polynomial, Khovanov complex, Khovanov homology, Reidemeister moves, virtual knot theory, differential, partial differential, grading, dotted grading, vector grading.
Based on a recently introduced by the author notion of {\em parity}, in the present paper we construct a sequence of invariants (indexed by natural numbers $m$) of long virtual knots, valued in certain simply-defined group ${\tilde G}_{m}$ (the Cayley graphs of these groups are represented by grids in the $(m+1)$-space); the conjugacy classes of elements of $G_{m}$ play the role of invariants of {\em compact} virtual knots. By construction, all invariants do not change under {\em virtualization}. Factoring the group algebra of the corresponding group by certain polynomial relations leads to finite order invariants of (long) knots which do not change under {\em virtualization
We construct a simple invariant of free link valued in a certain group by using parity.
The present paper is a review of the current state of Graph-Link Theory (graph-links are also closely related to homotopy classes of looped interlacement graphs), dealing with a generalisation of knots obtained by translating the Reidemeister moves for links into the language of intersection graphs of chord diagrams. In this paper we show how some methods of classical and virtual knot theory can be translated into the language of abstract graphs, and some theorems can be reproved and generalised to this graphical setting. We construct various invariants, prove certain minimality theorems and construct functorial mappings for graph-knots and graph-links. In this paper, we first show non-equivalence of some graph-links to virtual links.
We consider knot theories possessing a {\em parity}: each crossing is decreed {\em odd} or {\em even} according to some universal rule. If this rule satisfies some simple axioms concerning the behaviour under Reidemeister moves, this leads to a possibility of constructing new invariants and proving minimality and non-triviality theorems for knots from these classes, and constructing maps from knots to knots. Our main example is virtual knot theory and its simplifaction, {\em free knot theory}. By using Gauss diagrams, we show the existence of non-trivial free knots (counterexample to Turaev's conjecture), and construct simple and deep invariants made out of parity. Some invariants are valued in graph-like objects and some other are valued in groups. We discuss applications of parity to virtual knots and ways of extending well-known invariants. The existence of a non-trivial parity for classical knots remains an open problem.
By using parity arguments we prove that free knots are, generally, not invertible.
We investigate cobordisms of free knots. Free knots and links are also called homotopy classes of Gauss words and phrases. We define a new strong invariant of free knots which allows to detect free knots not cobordant to the trivial one.