Invariants And Pictures: Low-dimensional Topology And Combinatorial Group Theory

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Vassily Olegovich Manturov. Invariants And Pictures: Low-dimensional Topology And Combinatorial Group Theory

Invariants and Pictures

Preface

Acknowledgments

Contents

Chapter 1. Groups. Small Cancellations. Greendlinger Theorem

1.1Group diagrams language

1.1.1Preliminary examples

1.1.2The notion of a diagram of a group

1.1.3The van Kampen lemma

1.1.4Unoriented diagrams

1.2Small cancellation theory. 1.2.1Small cancellation conditions

1.2.2The Greendlinger theorem

1.3Algorithmic problems and the Dehn algorithm

1.4The Diamond lemma

Chapter 2. Braid Theory

2.1Definitions of the braid group

2.2The stable braid group and the pure braid group

2.3The curve algorithm for braids recognition

2.3.1Construction of the invariant

2.3.2Algebraic description of the invariant

2.4Virtual braids. Inclusion of classical braids into virtual braids

2.4.1Definitions of virtual braids

2.4.2Invariants of virtual braids

2.4.2.1A 2n-variable generalisation of the invariant

Chapter 3. Curves on Surfaces. Knots and Virtual Knots. 3.1Basic notions of knot theory

3.2Curve reduction on surfaces

3.2.1The disc flow

3.2.2Minimal curves in an annulus

3.2.3Proof of Theorems 3.3 and 3.4

3.2.4Operations on curves on a surface

3.3Links as braid closures. 3.3.1Classical case

3.3.2Virtual case

3.3.3An analogue of Markov’s theorem in the virtual case

Chapter 4. Two-dimensional Knots and Links

4.12-knots and links

4.2Surface knots

4.3Other types of 2-dimensional knotted surfaces

4.4Smoothing on 2-dimensional knots

4.4.1The notion of smoothing

4.4.2The smoothing process in terms of the framing change

4.4.3Generalised F-lemma

Chapter 5. Parity in Knot Theories. The Parity Bracket

5.1The Gaußian parity and the parity bracket

5.1.1The Gaußian parity

5.1.2Smoothings of knot diagrams

5.1.3The parity bracket invariant

5.1.4The bracket invariant with integer coefficients

5.2The parity axioms

5.3Parity in terms of category theory

5.4The L-invariant

5.5Parities on 2-knots and links

5.5.1The Gaußian parity

5.5.2General parity principle

5.6Parity Projection. Weak Parity. 5.6.1Gaußian parity and parity projection

5.6.2The notion of weak parity

5.6.3Functorial mapping f for Gaußian parity for free, flat and virtual knots

5.6.4The parity hierarchy on virtual knots

Chapter 6. Cobordisms. Applications of Parity Theory to Cobordisms of Knots

6.1Cobordism in knot theories. 6.1.1Basic definitions

6.1.2Cobordism types

6.2Sliceness criteria for certain families of framed graphs

6.2.1Odd framed graphs

6.2.2Iteratively odd framed graphs

6.2.3Multicomponent links

6.2.4Other results on free knot cobordisms

6.3L-invariant as an obstruction to sliceness

Chapter 7. General Theory of Invariants of Dynamical Systems and Groups

7.1Dynamical systems and their properties

7.2Free k-braids

7.3The main theorem

7.4Pictures

Chapter 8. Groups and Their Homomorphisms. Recognition of Free Braids. Explicit Examples

8.1Homomorphism of pure braids into

8.2 Homomorphism of pure braids into

8.3Homomorphism into a free group

8.4Free groups and crossing numbers

8.5Proof of Proposition 8.3

Chapter 9. Generalisations of the Groups. 9.1Indices from and Brunnian braids

9.2Groups with parity and points

9.2.1Connection between and

9.2.2Connection between and

9.3Parity for and invariants of pure braids

9.4Group with imaginary generators. 9.4.1Homomorphisms from classical braids to

9.4.2Homomorphisms from to

9.5-groups for simplicial complexes and the word problem on G2(K)

9.5.1-groups for simplicial complexes

9.5.2The word problem for G2(K)

Chapter 10. Representations of the Groups and Their Connections with Permutahedra. 10.1Faithful representation of Coxeter groups

10.1.1Coxeter group and its linear representation

10.1.2Faithful representation of Coxeter groups

10.2Groups and Coxeter groups C(n, 2)

Chapter 11. Realisation of Spaces with Action. The braid groups for higher-dimensional spaces

11.1Realisation of the groups

11.1.1Preliminary definitions

11.1.2The realisability of

11.1.3Constructing a braid from a word in

11.1.4The group Hk and the algebraic lemma

11.2Realisation of for n ≠ k + 1. Partial flag varieties

11.2.1A simple partial case

11.2.2General construction

11.3The -complex

Chapter 12. Word and Conjugacy Problems in Groups

12.1Conjugacy problem in

12.1.1The existence of the algorithmic solution of the conjugacy problem in the group

12.1.2Algorithm of solving the conjugacy problem in

12.2The word problem for

12.2.1Presentation of the group H4

12.2.2The Howie diagrams

12.2.3The solution to the word ‘problem in H4

Chapter 13. The Groups and Invariants of Manifolds

13.1Projective duality

13.2Embedded hypersurfaces

13.2.1Examples

13.3Immersed hypersurfaces

13.4Circles in 2-manifolds and the group

13.5Immersed curves in M2

13.6A map from knots to 2-knots

Chapter 14. Introduction

14.1The manifold of triangulations

Chapter 15. The Two-dimensional Case

15.1The group definition. A group homomorphism fromPBn

15.1.1Geometric description of the mapping from PBnto

15.1.2Algebraic description of the mapping from PBnto

15.2A group homomorphism from PBn to

15.2.1Geometric description of the mapping from PBnto

15.2.2Algebraic description of the mapping from PBnto

15.3A group homomorphism from PBn to

15.4Braids in3and groups

15.5Lines moving on the plane and the group

15.5.1A map from a group of good moving lines to

15.5.2A map from a group of good moving lines to

15.5.3A map from a group of good moving unit circles to

15.6A representation of braids via triangulations

15.7Decorated triangulations

Chapter 16. The Three-dimensional Case. 16.1The group

16.2The general strategy of defining for arbitrary k

16.3The groups

Chapter 17. Open Problems in the Groups and Theory and Related Fields. 17.1The groups and. 17.1.1Algeb raic ‘problems

17.1.2Topological problems

17.1.3Geometric problems

17.2G-braids

17.3Weavings

17.4Free knot cobordisms. 17.4.1Cobordism genera

17.5Picture calculus. 17.5.1Picture-valued solutions of the Yang–Baxter equations

17.5.2Picture-valued classical knot invariants

17.5.3Categorification of polynomial invariants

17.6Theory of secants

17.7Surface knots

17.7.1Parity for surface knots

17.8Link homotopy. 17.8.1Knots in Sg × S1

17.8.2Links in Sg × S1

17.8.3Degree of knots in Sg × S1

17.8.4Questions

Bibliography

Index

Отрывок из книги

Low-dimensional Topology and Combinatorial Group Theory

Invariants and Pictures

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11.Realisation of Spaces with Action

11.1Realisation of the groups .

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