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
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
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, . . . , mk+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
