Invariants and Pictures

The following sections are included:

• Preface
• Acknowledgments
• Contents

In the present chapter we discuss certain basic notions of combinatorial group theory. In particular, we recall the notion of small cancellation conditions for groups and study the diagrammatic method of describing such groups…

The present chapter is an introduction to the theory of braids, which has some nice intrinsically interesting properties. Here we highlight some ideas and definitions which will be important for our purposes later. In our overview of the basics of braid theory we will closely follow [Manturov, 2018].

The following sections are included:

• Basic notions of knot theory
• Curve reduction on surfaces
  • The disc flow
  • Minimal curves in an annulus
  • Proof of Theorems 3.3 and 3.4
  • Operations on curves on a surface
• Links as braid closures
  • Classical case
  • Virtual case
  • An analogue of Markov’s theorem in the virtual case

Similar to how classical 1-knots are embeddings of a circle into a 3-sphere, classical 2-knots are embeddings of a 2-sphere into ℝ⁴ (or a sphere S⁴), see Definition 4.1 later. The standard way of describing 2-knots is to consider their diagrams — two-dimensional complexes of a particular kind. A diagram is a generic projection of a knot in 4-space into a 3-subspace where for each double line it is said which sheet is “over” and which one is “under” and for every triple point the same is said for all three sheets incident to that point…

The foundation for parity theory was laid in the paper [Manturov, 2010] for one-dimensional (virtual) knots. The main idea behind it is: All crossings of a diagram of a (virtual) knot can be naturally split into even ones and odd ones so that this parity behaves nicely under Reidemeister moves. …

In this chapter we review an important and long-studied notion of cobordism of knots and extend it to the case of virtual knots and free knots. The latter case can be considered as graph cobordism. Those theories also can be studied from picture-valued standpoint. In particular, we give a sliceness (that is, null-cobordance) criterion for odd free knots which presents a finite algorithmic solution to the question whether an odd free knot is slice. This algorithm can be performed given a diagram of the knot. In that sense the diagram itself is the invariant of sliceness…

Usually, invariants of mathematical objects are valued in numerical or polynomial rings, rings of homology groups, etc. In the present chapter, we prove a general theorem about invariants of dynamical systems which are valued in groups very close to pictures, the so-called free k-braids…

In the present chapter, we work with groups $G_n^k$, which generalise classical braid groups and other groups in a very broad sense.

Perhaps, the easiest groups are free products of cyclic groups (finite or infinite). In these groups, the word problem and the conjugacy problem are solved extremely easily: we just contract a generator with its opposite until possible (in a word or in a cyclic word) and stop when it is impossible…

The following sections are included:

• Indices from $G_n^3$ and Brunnian braids
• Groups $G_n^2$ with parity and points
  • Connection between $G_{n,p}^2$ and $G_{n,d}^2$
  • Connection between $G_{n,d}^2$ and $G_{n+1}^2$
• Parity for $G_n^2$ and invariants of pure braids
• Group $G_n^3$ with imaginary generators
  • Homomorphisms from classical braids to ${\tilde{G}}_n^3$
  • Homomorphisms from ${\tilde{G}}_n^3$ to $G_{n+1}^3$
• The groups $G_n^k$ for simplicial complexes
  • $G_n^k$-groups for simplicial complexes
  • The word problem for G²(K)
• Tangent circles

The following sections are included:

• Faithful representation of Coxeter groups
  • Coxeter group and its linear representation
  • Faithful representation of Coxeter groups
• Groups $G_n^2$ and Coxeter groups C(n, 2)

The aim of the subsequent chapters is to construct spaces where $G_n^k$ (or a finite index subgroup of it) acts faithfully.

On this way, we shall define the braid groups for higher dimensional spaces (Part 4). This gives rise to a definition for braid groups for arbitrary Euclidean and real projective spaces — the objects for study of its own. Note that the braid groups for manifolds other than ℝⁿ, ℝPⁿ will be defined in Part 4…

In the present chapter we study the word and conjugacy problems in the groups $G_{k+1}^k$. Notice that, as was remarked in Section 11.1.4, the word problem in the group $G_4^3$ is solved by Theorem 11.5. So in the present chapter we concentrate on the conjugacy problem for $G_4^3$ and the word problem for $G_5^4$…

In the main principle formulated in Chapter 7, we spoke about the dynamics of n particles in a certain topological space.

In all examples considered in the previous chapters, all particles were points in certain topological spaces which led us to various configuration spaces. In the present chapter, we shall deal with submanifolds instead of points playing roles of particles, hence, with moduli spaces of such submanifolds…

The present part is the culmination of the whole book: here we construct “braid-group like” invariants of arbitrary manifolds.

Returning back to 1954, J.W. Milnor in his first paper on link group [Milnor, 1954] formulated the idea that manifolds which share lots of well known invariant (homotopy type, etc.) may have different link groups…

In Chapter 7 we defined a family of groups $G_n^k$ for two positive integers n > k, and formulated the following principle: If dynamical systems describing a motion of n particles admit some good codimension one property governed by exactly k particles, then these dynamical systems have a topological invariant valued in $G_n^k$.…

The following sections are included:

• The group ${\Gamma }_n^5$
• The general strategy of defining ${\Gamma }_n^k$ for arbitrary k
• The groups ${\tilde{\Gamma }}_n^k$

The following sections are included:

• The groups $G_n^k$ and ${\Gamma }_n^k$
  • Algebraic problems
  • Topological problems
  • Geometric problems
• G-braids
• Weavings
• Free knot cobordisms
  • Cobordism genera
• Picture calculus
  • Picture-valued solutions of the Yang–Baxter equations
  • Picture-valued classical knot invariants
  • Categorification of polynomial invariants
• Theory of secants
• Surface knots
  • Parity for surface knots
• Link homotopy
  • Knots in S_g × S¹
  • Links in S_g × S¹
  • Degree of knots in S_g × S¹
  • Questions

The following sections are included:

• Bibliography
• Index