Читать книгу Invariants And Pictures: Low-dimensional Topology And Combinatorial Group Theory - Vassily Olegovich Manturov - Страница 7
Contents
ОглавлениеIntroduction
1.Groups. Small Cancellations. Greendlinger Theorem
1.1.2The notion of a diagram of a group
1.2.1Small cancellation conditions
1.3Algorithmic problems and the Dehn algorithm
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
2.4.1Definitions of virtual braids
2.4.2Invariants of virtual braids
3.Curves on Surfaces. Knots and Virtual Knots
3.1Basic notions of knot theory
3.2Curve reduction on surfaces
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
3.3.1Classical case
3.3.3An analogue of Markov’s theorem in the virtual case
4.Two-dimensional Knots and Links
4.3Other types of 2-dimensional knotted surfaces
4.4Smoothing on 2-dimensional knots
4.4.2The smoothing process in terms of the framing change
Parity Theory
5.Parity in Knot Theories. The Parity Bracket
5.1The Gaußian parity and the parity bracket
5.1.2Smoothings of knot diagrams
5.1.3The parity bracket invariant
5.1.4The bracket invariant with integer coefficients
5.3Parity in terms of category theory
5.5Parities on 2-knots and links
5.6Parity Projection. Weak Parity
5.6.1Gaußian parity and parity projection
5.6.2The notion of weak parity
5.6.3Functorial mapping for Gaußian parity
5.6.4The parity hierarchy on virtual knots
6.1.1Basic definitions
6.2.2Iteratively odd framed graphs
6.2.4Other results on free knot cobordisms
6.3L-invariant as an obstruction to sliceness
The Groups
7.General Theory of Invariants of Dynamical Systems
7.1Dynamical systems and their properties
8.Groups and Their Homomorphisms
8.1Homomorphism of pure braids into
8.2Homomorphism of pure braids into
8.3Homomorphism into a free group
8.4Free groups and crossing numbers
9.Generalisations of the Groups
9.1Indices from and Brunnian braids
9.2Groups with parity and points
9.3Parity for and invariants of pure braids
9.4Group with imaginary generators
9.4.1Homomorphisms from classical braids to
9.5The groups for simplicial complexes
9.5.1-groups for simplicial complexes
9.5.2The word problem for G2(K)
10.Representations of the Groups
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)
11.Realisation of Spaces with Action
11.1Realisation of the groups .
11.1.3Constructing a braid from a word in
11.1.4The group Hk and the algebraic lemma
11.2Realisation of , n ≠ k + 1
12.Word and Conjugacy Problems in Groups
12.1.1Existence of the algorithmic solution
12.1.2Algorithm of solving the conjugacy problem in
12.2.1Presentation of the group H4
12.2.3The solution to the word problem in H4
13.The Groups and Invariants of Manifolds
13.4Circles in 2-manifolds and the group
13.6A map from knots to 2-knots
Manifolds of Triangulations
14.1The manifold of triangulations
15.2A group homomorphism from PBn to
15.3A group homomorphism from PBn to
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
16.1The group
16.2The general strategy of defining for arbitrary k
Unsolved Problems
17.1The groups and
17.1.1Algebraic problems
17.4.1Cobordism genera
17.5.1Picture-valued solutions of the Yang–Baxter equations
17.5.2Picture-valued classical knot invariants
17.5.3Categorification of polynomial invariants
17.7.1Parity for surface knots
17.8.1Knots in Sg × S1
