The study of knots and their properties is known as knot theory. Classical theory counts more than two hundred and fifty years. Knots appear in Chinese knotting, Tibetan Buddhism, intricate Celtic knot-work, in the 1200 year old Book of Kells, and so on. As a mathematical theory appeared in 1771 by French mathematician Vandermonde, when discussing the properties of knots related to the geometry of position. Mathematical studies of knots began in the 19th century with Gauss, who defined the linking integral. This theory was given its first impetus when Lord Kelvin proposed a theory that atoms were vortex loops, with different chemical elements consisting of different knotted configurations, P.G. Tait then cataloged possible loops with different knots by trial and error, the first knot tables with up to ten crossings, known as the Tait conjectures, this record motivated that knot theory became part of the emerging subject of topology. Over six billion knots and links have been tabulated since the beginnings of knot theory in the 19th century. Much progress has been made in the intervening years, for example, Alexander, Dehn, Klein, Reidemeister, and other outstanding mathematicians.

A knot is defined as a closed non-self-intersecting curve that is embedded in three dimensions and cannot be untangled to produce a simple loop (i.e., unknot). To a mathematician, an object is a knot only if its free ends are attached in some way so that the resulting structure consists of a single looped strand. A knot can be generalized to a link which is simply a knotted collection of one o more closed strands. A virtual knot represents a natural combinatorial generalization of a classical knot, simply introduced a new type of a crossing and extended new moves to the list of the Reidemester moves, the new crossing (virtual) should be treated as a diagrammatic picture of two part of a knot (a link) on the plane which are far from each other, and the intersection of these parts is artifact of such a drawing. In short, a virtual diagram (or a diagram of a virtual link) is the image of an inmersion of a framed 4-valent graph in R2 with a finite number of intersections of edges. Moreover, each intersection is a transverse double point which named a virtual crossing and mark by a small circle, and each vertex of the graph is endowed, with the classical crossing structure. The theory of knots in three-dimensional Euclidean space or in the three-sphere (the classical theory) is an integral part of a much larger theory, knots in 3-manifolds.

This remarkable book is the first systematic and full-length book on the theory of virtual knots and links, devoted to an intriguing and comprehensive study of (virtual and classical) knots as integral part. The book is self-contained. The mathematical material is sufficiently closed, and contains up-to-date exposition of the key aspect of virtual (and classical) knot theory. The book is quite accessible for undergraduate students of low courses, thus it can be used as a basic course book on virtual and classical knot theory. This book can also be useful for professionals and amateur mathematicians because it contains the newest and the most significant scientific developments in knot theory. The book was written using knots.tex fonts containing special symbols from knot Theory.

The aim of the present book is to describe the main concepts of modern knot theory together with full proofs that would be both accessible to beginners and useful for professionals. A large part of the present title is devoted to rapidly developing areas of modern knot theory: such as virtual knot theory and Legendrian knot theory.

Over the last decades, knot theory was enriched by numerous methods and subtle invariants, which today constitute a powerful tool in knot theory and low-dimensional topology. A breakthrough in knot theory is due to discoveries of Conway, Jones and Vassiliev (Conway and Jones polynomials, Vassiliev´s finite type invariants). For their impact to knot theory which related knot theory to various branches of mathematics and physics, Jones, Witten, Drinfeld (1990) y Kontsevich(1998) were awarded the highest honor of mathematics, The Fields medals.

Virtual knots were discovered by Louis H. Kauffman, in 1996, and independently by Goussarov, Polyak and Viro in 2000, about finite type invariants of classical and virtual knots. The first paper on virtual knot theory appeared in 1999, by Louis H. Kauffman in European Journal of Combinatory. The virtual knot theory helped one to understand better some aspects of the classical knot theory. By means of virtual knot theory, the problem of existence of combinatorial formulae for finite type invariants for classical knots was solved.

A common point of view allows us to treat classical and virtual knots uniformly.

A classical knot (link) can be given by a planar diagram. There are classical crossings and curves connecting crossings to each other. In the classical knot is possible that the curves connecting crossings can be chosen to be non-intersecting and in some cases it is impossible to situate these curves without additional intersections. This new intersections are marked as virtual (encircled) and we get a virtual diagram. Virtual crossings appear every time, when a 4-valent graph defined by classical crossings and ways of connection, is not planar, which happens quite often.

Thus, virtual knots are related to classical knots approximately in the same way as graphs are related to planar graphs. Herewith, the equivalence (isotopy) of classical knot diagrams is defined by means of formal combinatorial transformations (Reidemeister moves), which are applied to crossing lying close to each other.

From the topological point of view, the virtual knots are knots in thickened surfaces (products of sphere with handles with an interval) considered up to isotopy and stabilizations.

The book contains nine chapters, bibliography and an index, in 520 pp. It goes from the basics to the frontiers of research. The book is volume 51 of Series on Knots and Everything, polarized around the theory of Knot Theory. The questions treated reach out beyond theory itself into physics, mathematics, logic, linguistics, philosophy, biology and practical experienced. The book is dedicated to the memory of Oleg Vassilievich Manturov (1936-2011), father of Vassily Olegovich Manturov, first-named author of the book.

The first chapter, untitled basic, definitions and notions, devoted to the diagrammatic and combinatorial definitions and to a discovery of the self-linking number of a virtual knot. This chapter is a compendium, remarkable and fascinating encyclopedic treatment of basics.

In Chapter 2, is dedicated to virtual knots and the three dimensional topology. The authors present a discussion of Kuperberger´s Theorem, and the surface genus of virtual knots. Kuperberger proved that if a virtual knot (link) is represented in its minimal genus surface, then this embedding type is unique. The result is interesting because is fruitful for getting deeper invariants, a stronger version of the Jones polynomial for virtual knots and links. This chapter also proves that virtual knots are algorithmically recognizable by generalizing the technique of Haken and Hemion for classical knots. The recognition problem for classical knots was one of the central problems in low-dimensional topology. Its first solution is related to Haken´s normal surface theory, and the final steps belong to Matveev. The result of algorithmic recognizability of a certain object in low-dimensional topology is important, because, in low-dimensional topology, the algorithmic non-realizability takes place for many objects. When virtual knot theory appeared, the problem of algorithmic realizability of virtual knots arose. This problem is resolved positively, in this chapter. This result relies not only on Haken´s theory, but also on Kuperberg´s theorem.

In chapter 3 on consider quandles (distributive groupoids) and their generalizations to virtual knots. Generalizations of invariants that emerged, from virtual knot theory, is the use of the biquandle, and its powerful applications to the theory. Biquandle is a generalization of the quandle, which in turn generalizes the fundamental group of a knot (link). It worth the Lie algebraic techniques for invariants, long virtual knots, and the hierarchy of virtual knots, where have ordered by some choice of an ordinal, in terms of their ability to move across one another. Some invariants are found here by the hierarchy of virtual knots.

Chapter 4 does the basics of the Jones-Kauffman polynomials via the bracket state sum and introduces the concept of atoms (surfaces, orientable or not, bearing the virtual knot diagram). It also treated the chord diagrams and the passage from atoms to chord diagrams. The spanning tree, the leading and lowest terms of the Kauffman bracket polynomial, Kauffman bracket for rigid knots, and so on. It ends with the study of minimal diagrams of long virtual knots.

In Chapter 5, in the present book, on dealing with many ideas related to virtual theory can be directed to the study of Khovanov homology, as the homology of an algebraic complex which is constructed with a diagram of a knot (link), this homology detects the unknot, which seriously enlarges horizons of the theory. The Khovanov theory, associates with each knot diagram a chain complex, whose homology is a knot invariant, and the Euler characteristic of this complex coincides with the Jones polynomial. These chains constructed with a knot diagram, correspond to formal smoothing of this diagram at all classical crossings. It is interesting that the differentials of the Khovanov complex are defined combinatorially, and the homology is invariant under Reidemeister moves. To extend Khovanov homology theory to virtual knots, Manturov had to revisit completely the theory and construct a complex homotopically equivalent to the usual Khovanov complex. The problem, in the case of virtual knots, is related to the well definedness in order for the square of the differential to be equal to zero. This procedure needs to be computed, and compared with the Khovanov Rozansky Categorified Link Homology. A key role in this construction was played by the notion of atom and played a crucial role in the proof of Vassiliev´s conjecture. Such construction is very nice because required a number of new ideas: orientation and enumeration of the state curves; twisted coefficients in the Frobenius algebra representing, and the usage of exterior products instead of usual products. In short, the authors gave a first combinatorial solution to the question of constructing integral Khovanov homology for the virtual knots and links.

Chapter 6 deals with virtual braids and the work of Kamada (2007), and Kauffman and Lambropoulou(2006), and the considerations of the invariants of virtual braids, in the following cases, the construction of the main invariant, then the representation of virtual braid group. On the other hand, studies on completeness in the classical case and the case of two-strand braids.

Chapter 7 treats combinatorial aspects of the Vassieliev invariant theory and the work of Goussarov, Polyak and Viro who used virtual knots in the guise of general Gauss diagrams to construct a theory of Gauss diagram formulas for virtual knots. Virtual knot theory, its constructions and methods are closely related to various branches of classical knot theory, in particular, to Vassiliev invariants. These occupy a special position in classical knot theory; it turned out just, when this theory appeared, that all polynomial and quantum invariants were expressible in terms of Vassiliev invariants. In the case of virtual knots, the theory of Vassiliev knot invariants is much more complicated; even the space of order zero invariants is infinite-dimensional. In this chapter, by using atoms and d-diagrams, on proved Vassiliev´s conjecture about planarity of framed 4.valent graphs (graphs where at each vertex four half-edges are split into two pairs of opposite ones); this conjecture solved positively, plays a key role in Vassiliev´s work on the existence of integer-valued combinatorial formulae for invariants of finite order.

The chapter 8 is devoted to parity in knot theory. In virtual knot theory, there are many unexpected invariants which do not take place in the classical case. This chapter includes work on the Goldman bracket, and the Turaev cobracter and on cobordism of free knots. At first it was thought (a conjecture of Turaev) that free knots were trivial, but the most striking example of such theory is the parity theory conceived by Vassily Olegovich Manturov, where all classical crossings are either even or odd, herewith the property of being even is naturally preserve by Reidemeister moves. By parity, we mean any such natural way of labeling of all classical crossing which is defines for all knots from this theory. Manturov showed, using parity that this is not the case and that there are non-trivial cobordism classes of free knots. By means of parity, one can construct functorial mappings from knots to knots, filtrations on the space of knots, refine many invariants and prove minimality of many series of knot diagrams. The analogous virtual knot theory is to study virtual knots up to change of orientation of the crossings as we describe of the beginnings, to study virtual knots up to virtualization equivalence. The existence of different parities and different projections (from knots to knots) allows one to establish various filtrations on the space of knots. Besides that, such projections allows, one to lift invariants, from classical knots to virtual knots.

Finally the Chapter 9 on Graph link theory, a further combinatorial. The passage from classical knots to virtual knots can also be motivated by representing Reidemeister moves in the language of Gauss diagrams. Every Gauss diagram is a circle with a collection of pairs of points (all points mutually disjoint); every pair of points is endowed with an arrow from one point to the other and a sign. Each chord diagram of such sort has an intersection graph. Vertices of the intersection graph correspond to the chords, and two vertices are adjacent whenever the corresponding chords are linked. To such graphs, one can extend Reidemeister moves. Note that not all simple (without multiple edges and loops) graphs originate from chord diagrams. When passing from intersection graphs of chord diagrams to arbitrary graphs and extending Reidemeister moves to such graphs, we end up with the graph-link theory due to the authors of the present book. An analogous theory was constructed by Traldi and Zulli. Graph link can be treated as diagramless knot theory. Such links have crossing, but they do not have arcs connecting this crossings since the corresponding graphs are not intersection graphs of any chord diagrams and thus they are not drawable on the plane. It turns out, however, that to graph-links one can extend many methods of the classical and virtual knot theory, in particular, the parity theory. We have constructed various invariants, proved the equivalence of two approaches to graph-knots: the one suggested by the authors and the one suggested by Traldi and Zulli. We have constructed various invariants showing non-realizability of graph-links (the fact that a graph-link has no drawable representative). A remarkable achievement in the graph-link theory is the work by Nikonov, who constructed Khovanov homology theory for graph-links with coefficients from Z2. Unlike the usual Kauffman bracket when one had to count the number of non-existing state-circles, for this problem one had to understand how these non-existing circles might interfere in order to construct the differential in the Khovanov complex.

The theories mentioned above are related to different problems of combinatorics, three-dimensional topology, and four-dimensional topology, representation theory for Lie groups and algebras. Representation theory is the starting point for constructing quantum invariants of knots and 3- manifolds.

The book is the result of the research for over 10 years, different questions of virtual knot theory were discussed in the seminar “Knots and the representation theory” and Seminar on “Tensor and Vector Analysis” (the latter exists since 1920s) in the Moscow State University.

Definitely the book is high recommended to undergraduate, graduate, professionals and amateur mathematicians, because it goes from the basics to the frontiers of research. Finally, as L.H. Kauffman says: “this book self-contained is motivated, to delve, into the adventure proposed, by this intriguing and remarkable book”.