Читать книгу Invariants And Pictures: Low-dimensional Topology And Combinatorial Group Theory - Vassily Olegovich Manturov - Страница 5

A long time ago, when I first encountered knot tables and started unknotting knots “by hand”, I was quite excited with the fact that some knots may have more than one minimal representative. In other words, in order to make an object simpler, one should first make it more complicated. For example, see Fig. 0.1 [Kauffman and Lambropoulou, 2012]: this diagram represents the trivial knot, but in order to simplify it, one needs to perform an increasing Reidemeister move first.

Fig. 0.1Culprit knot

Being a first year undergraduate student (in Moscow State University), I first met free groups and their presentation. The power and beauty, and simplicity of these groups for me were their exponential growth and extremely easy solution to the word problem and conjugacy problem by means of a gradient descent algorithm in a word (in a cyclic word, respectively).

Also, I was excited with the Diamond lemma: a simple condition which guarantees the uniqueness of the minimal objects, and hence, solution to many problems (Chapter 1.4).

Being a last year undergraduate and teaching a knot theory course for the first time, I thought: “Why do not we have it (at least partially) in knot theory?”

Fig. 0.2The Diamond lemma

By that time I knew about the Diamond lemma and solvability of many problems like word problem in groups by gradient descent algorithm. The van Kampen lemma and Greendlinger’s theorem came to my knowledge much later.

I spent a lot of time working with virtual knot theory; my doctoral (habilitation) thesis [Manturov, 2007] was devoted to various problems in that theory: from algorithmic recognition of virtual knots to the construction of Khovanov homology for virtual knots with arbitrary coefficients.

Virtual knot theory (Chapter 3), which can be formally defined via Gauß diagrams which are not necessarily planar, is a theory about knots in thickened surfaces Sg × I considered up to addition/removal of nugatory handles. It contains classical knot theory as a proper part: classical knots can be thought of as knots in the thickened sphere. Hence, virtual knots have a lot of additional information coming from the topology of the ambient space (Sg).

By playing with formal Gauß diagrams, I decided to drop any arrow and sign information and called such objects free knots [Manturov, 2009].

It was an interesting puzzle for me in December 2008 in Heidelberg to construct invariants of free knots as I had never heard of any. What one should pay attention to is that all chords of a Gauß diagram of a virtual knots can be odd and even. Gauß himself knew that Gauß diagrams of planar curves and knots have no odd chords. Hence, odd chords can be the key point of non-triviality and non-classicality. When looking at Reidemeister moves, one can see that the chord taking part in a first Reidemeister move is even; two chords taking part in a second Reidemeister move are of the same parity, and the sum of parities of the three chords taking part in a third Reidemeister move is 0 modulo 2 if we count odd chords as 1 and even chords as 0 (Definition 5.8). Hence, odd chords can only cancel with “neighbouring” odd chords by the second Reidemeister moves; otherwise they persist.

In January 2009, I constructed a state-sum invariant of free knots valued in diagrams of free knots, i.e., framed four-valent graphs¹ [Manturov, 2010]. For states, I was taking all possible smoothings at even crossings, imposing some diagrams to be zero. This invariant was constructed in such a way that all odd chords persisted and I got the formula

whenever K is a diagram of a free knot with all chords where no two chords can be cancelled by a second Reidemeister move (see Section 5.1).

The deep sense of this formula can be expressed as follows:

If a virtual diagram is complicated enough, then it realises itself.

Namely, K on the left hand side is some free knot diagram K, and K on the right hand side is a concrete graph. Hence, if K′ is another diagram of the same knot, we shall have [K′] = K meaning that K is obtained as a result of smoothing of K′. This principle is depicted on Fig. 0.3.

Fig. 0.3A picture which is its own invariant

This is an example of what we miss in classical knot theory: local minimality yields global minimality or, in other words, if a diagram is odd and irreducible then it is minimal in a very strong sense (Lemma 5.1, Section 5.1). Not only one can say that K′ has larger crossing number than K, we can say that K “lives” inside K′. Having these “graphical” invariants, we get immediate consequences about many characteristics of K. The main problem is to construct invariants of similar nature in the case of classical knot theory.

Something similar can be seen in other situations: free groups or free products of cyclic groups, cobordism theory for free knots, or while considering other geometrical problems. For example, if we want to understand the genus of a surface where the knot K can be realised, it suffices for us to look at the minimal genus where the concrete diagram K can be realised for the genus of any other K′ is a priori larger than that of K [Manturov, 2012b], see also [Ilyutko, Manturov and Nikonov, 2011].

Actually, having lectured knot theory over many years and having published several knot theory books by that time [Manturov, 2018; Ilyutko and Manturov, 2013; Ilyutko, Manturov and Nikonov, 2011], I knew that a knot group (fundamental group of the complement to a knot) may be very complicated, hence, it may contain powerful information inside.

What is the main difference between classical knot theory and virtual knot theory? In my opinion, the existence of a large ambient group (π₁(Sg)) in the virtual knot case. By extracting some information out of it, one can get parities and other enhancements for knots, hence, leading us to the above “picture-valued” invariants (Section 5.1).

But how to extract nice group information out of classical knots? For my purposes (say, for picture-valued brackets), there is a lack of “canonical coordinate system” how to make the knot see these nodes (crossings) visible in a way to get some parity.

As a topologist, I was always interested in cobordisms and concordance: much more subtle equivalence relations than isotopy or homotopy. As a knot theorist, I knew from my childhood about the Fox conjecture: all slice knots are conjectured to be ribbon.

Now, let us look at free knots: very coarse objects; it looks like if we impose a coarse equivalence relation, like cobordism, it will kill everything. However, it turned out not only that cobordism classes are non trivial, but for odd irreducible free knots the Fox conjecture is true (it is my joint work with D. A. Fedoseev [Fedoseev and Manturov, 2019b; Fedoseev and Manturov, 2018]).

Again we can say that some local information allows one to judge about some global dynamics: if it is not possible to pair chord ends and cap an odd framed 4-graph at once without singularities, just with double lines, then there is no chance for it to be capped after any long sequence of Reidemeister moves, maxima and minima, triple points and cusps (statics).

This takes me back to the time of my habilitation thesis. Once writing a knot theory paper and discussing it with Oleg Yanovich Viro, I wrote “a classical knot is an equivalence class of classical knot diagrams modulo Reidemeister moves”. Well, — said Viro, — you are restricting yourself very much. Staying at this position, how can you prove that a non-trivial knot has a quadrisecant?

Reflecting this after several years, I understood that it is not quite necessary to consider knots by using Reidemeister planar projections, one can look for other “nodes”.

This led me to my initial preprint [Manturov, 2015a] and to an extensive study of braids and dynamical systems. This happened around New Year 2015.

Namely, I looked at usual Artin braids as everybody does: as dynamical systems of points in ², but, instead of creating Reidemeister’s diagram by projecting braids to a screen (say, the plane Oxz), I decided to look at those moments when some three points are collinear. This is quite a good property of a “node” which behaves nicely under generic isotopy. Let us denote such situations by letters aijk where i, j, k are numbers of points (this triple of numbers is unordered).

When considering four collinear points, we see that the tetrahedron (Zamolodchikov, see, for example, [Etingof, Frenkel and Kirillov, 1998]) equation emerges. Namely, having a dynamics with a quadruple point and slightly perturbing it, we get a dynamics, where this quadruple point splits into four triple points.

Writing it algebraically, we get:

Definition 0.1. The groups are defined as follows.

where the generators am are indexed by all k-element subsets of {1, . . . , n}, the relation (1) means

(2) means

and, finally, the relation (3) looks as follows. For every set U ⊂ {1, . . . , n} of cardinality (k + 1), let us order all its k-element subsets arbitrarily and denote them by m¹, . . . , m^k+1. Then (3) is:

This situation with the Zamolodchikov equation happens almost everywhere, hence, I formulated the following principle:

If dynamical systems describing the motion of n particles possess a nice codimension one property depending on exactly k particles, then these dynamical systems admit a topological invariant valued in .

In topological language, it means that we get a certain homomorphism from some fundamental group of a topological space to the groups .

Collecting all results about the groups, I taught a half-year course of lectures in the Moscow State University entitled “Invariants and Pictures” and a 2-week course in Guangzhou. The notes taken by my colleagues I. M. Nikonov², D. A. Fedoseev and S. Kim were the starting point for the present book.

Since that time, my seminar in Moscow, my students and colleagues in Moscow, Novosibirsk, Beijing, Guangzhou, and Singapore started to study the groups , mostly from two points of view:

From the topological point of view, which spaces can we study?

Besides the homomorphisms from the pure braid group PBn to and and (Sections 8.1 and 8.2), I just mention that I invented braids for higher-dimensional spaces (or projective spaces).

Of course, the configuration space C(^k−1, n) is simply connected for k > 3 but if we take some restricted configuration space C′(^k−1, n), it will not be simply connected any more and leads to a meaningful notion of higher dimensional braid (Chapter 11).

What sort of the restriction do we impose? On the plane, we consider just braids, so we say that no two points coincide. In ³ we forbid collinear triples, in ⁴ we forbid coplanar quadruple of points.

When I showed the spaces I study to my coauthor, Jie Wu, he said: look, these are k-regular embeddings, they go back to Carol Borsuk. Indeed, after looking at some papers by Borsuk, I saw similar ideas were due to P.L. Chebysheff ([Borsuk, 1957; Kolmogorov, 1948]).

By the way, once Wu looked at the group , he immediately asked about the existence of simplicial group structure on such groups, the joint project we are working on now with S. Kim, J. Wu, F. Li.

An interested reader may ask whether such braids exist not only for ^k (or Pk), but also for other spaces. This question we shall touch on later.

From the algebraic point of view, why are these groups good, how are they related to other groups, how to solve the word and the conjugacy problems, etc.?

It is impossible to describe all directions of the group theory in the preface, the reader will find many directions in the unsolved problem list; I just mention some of them.

For properties of , we can think of them as n-strand braids with k-fold strand intersection.

Like , there are nice “strand forgetting” and “strand deletion” maps to and , see Fig. 0.4.

The groups have lots of epimorphisms onto free products of cyclic groups; hence, invariants constructed from them are powerful enough and easy to compare.

For example, the groups are commensurable with some Coxeter groups of special type, see Fig. 0.5, which immediately solves the word problem for them.

As Diamond lemma works for Coxeter groups, it works for , and in many other places throughout the book.

After a couple of years of study of , I understood that I was not completely free and this approach is still somewhat restrictive. Well, we can study braids, we can invent braids in ⁿ and Pn, but what if we consider just braids on a 2-surface? What can we study then? The property “three points belong to the same line” is not quite good even in the metrical case because even if we have a Riemannian metric on a 2-surface of genus g, there may be infinitely many geodesics passing through two points. Irrational cables may destroy the whole construction.

Then I decided to transform the “-point of view” to make it more local and more topological. Assume we have a collection of points in a 2-surface and seek -property: four points belong to the same circle.

Consider a 2-surface of genus g with N points on it. We choose N to be sufficiently large and put points in a position to form the centers of Voronoï cells. It is always possible for the sphere g = 0, and for the plane we may think that all our points live inside a triangle forming a Voronoï tiling of the latter.

Fig. 0.4Maps from to and

We are interested in those moments when the combinatorics of the Voronoï tiling changes, see Fig. 0.7.

This corresponds to a flip, the situation when four nearest points belong to the same circle. This means that no other point lies inside the circle passing through these four, see [Gelfand, Kapranov and Zelevinsky, 1994].

The most interesting situation of codimension 2 corresponds to five points belonging to the same circle.

Fig. 0.5The Cayley graph of the group and the Coxeter group C(3, 2)

Fig. 0.6Flips on a pentagon

This leads to the relation:

Note that unlike the case of , here we have five terms, not ten. What is the crucial difference? The point is that if we have five points in the neighbourhood of a circle, then every quadruple of them appears to be on the same circle twice, but one time the fifth point is outside the circle, and one time it is inside the circle. We denote the set {1, . . . , n} by n and introduce the following

Definition 0.2. The group is the group given by group presentation generated by subject to the following relations:

Fig. 0.7Voronoï tiling change

(1),

(2),

(3) for distinct i, j, k, l, m,

(4) for distinct i, j, k, l, m.

Just like we formulated the principle, here we could formulate the principle in whole generality, but we restrict ourselves with several examples.

It turns out that groups have nice presentation coming from the Ptolemy relation and the cluster algebras. The Ptolemy relation

says that the product of diagonals of an inscribed quadrilateral equals the sum of products of its opposite faces, see Fig. 0.8.

Fig. 0.8The Ptolemy relation

We can use it when considering triangulations of a given surface: when performing a flip, we replace one diagonal (x) with the other diagonal (y) by using this relation. It is known that if we consider all five triangulations of the pentagon and perform five flips all the way around, we return to the initial triangulation with the same label, see Fig. 0.6.

This well known fact gives rise to presentations of .

Thus, by analysing the groups , we can get

(1)Invariants of braids on 2-surfaces valued in polytopes;

(2)Invariants of knots;

(3)Relations to groups ;

(4)Braids on ³.

Going slightly beyond, we can investigate braids in ³ and the configuration space of polytopes.

We will not say much about the groups for k > 4. The main idea is:

(1)Generators (codimension 1) correspond to simplicial (k − 2)-polytopes with k vertices;

(2)The most interesting relations (codimension 2) correspond to (k − 2)-polytopes with k + 1 vertices.

It would be extremely interesting to establish the connection between with the Manin–Schechtmann “higher braid groups” [Manin and Schechtmann, 1990], where the authors study the fundamental group of complements to some configurations of complex hyperplanes.

It is also worth mentioning, that the relations in the group resemble the relations in Kirillov–Fomin algebras, see [Fomin and Kirillov, 1999]. For that reason it seems interesting to study the interconnections between those objects.

Finally, our invariants may not be just group-valued: some variations of admit simplicial group structures, which is studied now in a joint work with S. Kim, F. Li and J. Wu.

Note that the present book is very much open-ended. On one hand, the invariants of manifolds are calculated in some explicit cases. On the other hand, one can vary “the -principle” and “-principle” together with the groups itself and try to find invariants of manifolds depending on complex structures, spin-structures, and other structures by looking at some “good codimension 1 properties” and creating interesting configuration spaces.

The present book has the following structure. In Part 1 we review basic notions of knot theory and combinatorial group theory: groups and their presentations, van Kampen diagrams, braid theory, knot theories and the theory of 2-dimensional knots. Part 2 is devoted to the parity theory and its applications to cobordisms of knots and free knots. In Part 3 we present the theory of groups and their relations to invariants of dynamical systems. Part 4 deals with the notion of manifold of triangulations, higher dimensional braids, and investigates the groups . In the final Part 5 we present a list of unsolved problems in the theories discussed in the present book.

Vassily Olegovich Manturov 2019

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¹After I constructed such invariant, I learnt that free knots were invented by Turaev five years before that and thought to be trivial [Turaev, 2007], hence, I disproved Turaev’s conjecture without knowing that.

²I.M. Nikonov coauthored my first published paper about