Invariants and pictures : low-dimensional topology and combinatorial group theory

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Contents Preface Acknowledgments Introduction 1. Groups. Small Cancellations. Greendlinger Theorem 1.1 Group diagrams language 1.1.1 Preliminary examples 1.1.2 The notion of a diagram of a group 1.1.3 The van Kampen lemma 1.1.4 Unoriented diagrams 1.2 Small cancellation theory 1.2.1 Small cancellation conditions 1.2.2 The Greendlinger theorem 1.3 Algorithmic problems and the Dehn algorithm 1.4 The Diamond lemma 2. Braid Theory 2.1 Definitions of the braid group 2.2 The stable braid group and the pure braid group 2.3 The curve algorithm for braids recognition 2.3.1 Construction of the invariant 2.3.2 Algebraic description of the invariant 2.4 Virtual braids. Inclusion of classical braids into virtual braids 2.4.1 Definitions of virtual braids 2.4.2 Invariants of virtual braids 2.4.2.1 A 2n-variable generalisation of the invariant 3. Curves on Surfaces. Knots and Virtual Knots 3.1 Basic notions of knot theory 3.2 Curve reduction on surfaces 3.2.1 The disc flow 3.2.2 Minimal curves in an annulus 3.2.3 Proof of Theorems 3.3 and 3.4 3.2.4 Operations on curves on a surface 3.3 Links as braid closures 3.3.1 Classical case 3.3.2 Virtual case 3.3.3 An analogue of Markov's theorem in the virtual case 4. Two-dimensional Knots and Links 4.1 2-knots and links 4.2 Surface knots 4.3 Other types of 2-dimensional knotted surfaces 4.4 Smoothing on 2-dimensional knots 4.4.1 The notion of smoothing 4.4.2 The smoothing process in terms of the framing change 4.4.3 Generalised F-lemma Parity Theory 5. Parity in Knot Theories. The Parity Bracket 5.1 The Gauian parity and the parity bracket 5.1.1 The Gaubian parity 5.1.2 Smoothings of knot diagrams 5.1.3 The parity bracket invariant 5.1.4 The bracket invariant with integer coefficients 5.2 The parity axioms 5.3 Parity in terms of category theory 5.4 The L-invariant 5.5 Parities on 2-knots and links 5.5.1 The Gaubian parity 5.5.2 General parity principle 5.6 Parity Projection. Weak Parity 5.6.1 Gaubian parity and parity projection 5.6.2 The notion of weak parity 5.6.3 Functorial mapping f for Gaubian parity for free, flat and virtual knots 5.6.4 The parity hierarchy on virtual knots 6. Cobordisms 6.1 Cobordism in knot theories 6.1.1 Basic definitions 6.1.2 Cobordism types 6.2 Sliceness criteria for certain families of framed graphs 6.2.1 Odd framed graphs 6.2.2 Iteratively odd framed graphs 6.2.3 Multicomponent links 6.2.4 Other results on free knot cobordisms 6.3 L-invariant as an obstruction to sliceness The Groups Gkn 7. General Theory of Invariants of Dynamical Systems 7.1 Dynamical systems and their properties 7.2 Free k-braids 7.3 The main theorem 7.4 Pictures 8. Groups Gkn and Their Homomorphisms 8.1 Homomorphism of pure braids into G3n 8.2 Homomorphism of pure braids into G4n 8.3 Homomorphism into a free group 8.4 Free groups and crossing numbers 8.5 Proof of Proposition 8.3 9. Generalisations of the Groups Gkn 9.1 Indices from G3n and Brunnian braids 9.2 Groups G2n with parity and points 9.2.1 Connection between G2n;p and G2n;d 9.2.2 Connection between G2n;d and G2n+1 9.3 Parity for G2n and invariants of pure braids 9.4 Group G3n with imaginary generators 9.4.1 Homomorphisms from classical braids to eG3n 9.4.2 Homomorphisms from eG3n to G3n+1 9.5 Gkn-groups for simplicial complexes and the word problem on G2(K) 9.5.1 Gkn-groups for simplicial complexes 9.5.2 The word problem for G2(K) 9.6 Yet another braid invariant: codimension one properties arising from tangent circles 10. Representations of the Groups Gkn 10.1 Faithful representation of Coxeter groups 10.1.1 Coxeter group and its linear representation 10.1.2 Faithful representation of Coxeter groups 10.2 Groups G2n and Coxeter groups C(n; 2) 11. Realisation of Spaces with Gkn Action 11.1 Realisation of the groups Gkk+1 11.1.1 Preliminary definitions 11.1.2 The realisability of Gkk+1 11.1.3 Constructing a braid from a word in Gkk+1 11.1.4 The group Hk and the algebraic lemma 11.2 Realisation of Gkn for n 6= k + 1. Partial ag varieties 11.2.1 A simple partial case 11.2.2 General construction 11.3 The Gkn-complex 12. Word and Conjugacy Problems in Gkk+1 Groups 12.1 Conjugacy problem in G34 12.1.1 The existence of the algorithmic solution of the conjugacy problem in the group G34 12.1.2 Algorithm of solving the conjugacy problem in G34 12.2 The word problem for G45 12.2.1 Presentation of the group H4 12.2.2 The Howie diagrams 12.2.3 The solution to the word problem in H4 13. The Groups Gkn and Invariants of Manifolds 13.1 Projective duality 13.2 Embedded hypersurfaces 13.2.1 Examples 13.3 Immersed hypersurfaces 13.4 Circles in 2-manifolds and the group G3n 13.5 Immersed curves in M2 13.6 A map from knots to 2-knots Manifolds of Triangulations 14. Introduction 14.1 The manifold of triangulations 15. The Two-dimensional Case 15.1 The group 4n definition. A group homomorphismfrom PBn to 4n 15.1.1 Geometric description of the mapping from PBn to 4n. 15.1.2 Algebraic description of the mapping from PBn to 4n. 15.2 A group homomorphism fr om PBn to 4n x 4n 15.2.1 Geometric description of the mapping from PBn to 4n x 4n. 15.2.2 Algebraic description of the mapping from PBn to 4n x 4n. 15.3 A group homomorphism from PBn to 4n x ... x 4n 15.4 Braids in R3 and groups 4n 15.5 Lines moving on the plane and the group kn 15.5.1 A map from a group of good moving lines to G4n 15.5.2 A map from a group of good moving lines to 4n 15.5.3 A map from a group of good moving unit circles to 4n 15.6 A representation of braids via triangulations 15.7 Decorated triangulations 16. The Three-dimensional Case 16.1 The group 5n 16.2 The general strategy of defining kn for arbitrary k 16.3 The groups ~ kn Unsolved Problems 17. Open Problems 17.1 The groups Gkn and kn 17.1.1 Algebraic problems 17.1.2 Topological problems 17.1.3 Geometric problems 17.2 G-braids 17.3 Weavings 17.4 Free knot cobordisms 17.4.1 Cobordism genera 17.5 Picture calculus 17.5.1 Picture-valued solutions of the Yang{Baxter equations 17.5.2 Picture-valued classical knot invariants 17.5.3 Categorification of polynomial invariants 17.6 Theory of secants 17.7 Surface knots 17.7.1 Parity for surface knots 17.8 Link homotopy 17.8.1 Knots in Sg x S1 17.8.2 Links in Sg x S1 17.8.3 Degree of knots in Sg x S1 17.8.4 Questions Bibliography Index