The book is the first systematic research completely devoted to a comprehensive study of virtual knots and classical knots as its integral part. The book is self-contained and contains up-to-date exposition of the key aspects of virtual (and classical) knot theory. Virtual knots were discovered by Louis Kauffman in 1996. When virtual knot theory arose, it became clear that classical knot theory was a small integral part of a larger theory, and studying properties of virtual knots helped one understand better some aspects of classical knot theory and encouraged the study of further problems. Virtual knot theory finds its applications in classical knot theory. Virtual knot theory occupies an intermediate position between the theory of knots in arbitrary three-manifold and classical knot theory. In this book we present the latest achievements in virtual knot theory including Khovanov homology theory and parity theory due to V.O. Manturov and graph-link theory due to both authors. 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. Graph-links can be treated as """"diagramless knot theory"""": such """"links"""" have crossings, but they do not have arcs connecting these crossings. It turns out, however, that to graph-links one can extend many methods of classical and virtual knot theories, in particular, the Khovanov homology and the parity theory.

Description-Table Of Contents

1. Basic definitions and notions. 1.1. Classical knots. 1.2. Virtual knots. 1.3. Self-linking number -- 2. Virtual knots and three-dimensional topology. 2.1. Introduction. 2.2. The Kuperberg theorem. 2.3. Genus of a virtual knot. 2.4. Recognition of virtual links -- 3. Quandles (distributive groupoids) in virtual knot theory. 3.1. Introduction. 3.2. Quandles and their generalizations. 3.3. Long virtual knots. 3.4. Virtual knots and infinite-dimensional Lie algebras. 3.5. Hierarchy of virtual knots -- 4. The Jones–Kauffman polynomial: atoms. 4.1. Introduction. 4.2. Basic definitions. 4.3. The polynomial [symbol]: minimality problems. 4.4. Rigid virtual knots. 4.5. Minimal diagrams of long virtual knots -- 5. Khovanov homology. 5.1. Introduction. 5.2. Basic constructions: the Jones polynomial [symbol]. 5.3. Khovanov homology with [symbol]-coefficients. 5.4. Khovanov homology of double knots. 5.5. Khovanov homology and atoms. 5.6. Khovanov homology and parity. 5.7. Khovanov homology for virtual links. 5.8. Spanning tree for Khovanov complex. 5.9. The Khovanov polynomial and Frobenius extensions. 5.10. Minimal diagrams of links -- 6. Virtual braids. 6.1. Introduction. 6.2. Definitions of virtual braids. 6.3. Virtual braids and virtual knots. 6.4. The Kauffman bracket polynomial for braids. 6.5. Invariants of virtual braids -- 6. Vassiliev's invariants and framed graphs. 7.1. Introduction. 7.2. The Vassiliev invariants of classical knots and J-invariants of curves. 7.3. The Goussarov–Polyak–Viro approach. 7.4. The Kauffman approach. 7.5. Vassiliev's invariants coming from the invariant [symbol]. 7.6. Infinity of the number of long virtual knots. 7.7. Graphs, chord diagrams and the Kauffman polynomial. 7.8. Euler tours, Gauss circuits and rotating circuits. 7.9. A proof of Vassiliev's conjecture. 7.10. Embeddings of framed 4-graphs -- 8. Parity in knot theory: free-knots: cobordisms. 8.1. Introduction. 8.2. Free knots and parity. 8.3. A functorial mapping f. 8.4. Invariants. 8.5. Goldman's bracket and Turaev's cobracket. 8.6. Applications of Turaev's Delta. 8.7. An analogue of the Kauffman bracket. 8.8. Virtual crossing numbers for virtual knots. 8.9. Cobordisms of free knots -- 9. Theory of graph-links. 9.1. Introduction. 9.2. Graph-links and looped graphs. 9.3. Parity, minimality and non-trivial examples. 9.4. A generalization of Kauffman's bracket and other invariants. Minimality theorems.