Maps from knots in the cylinder to at-virtual knots V.O.Manturov I.M.Nikonov October 19 2022

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Maps from knots in the cylinder to ﬂat-virtual knots

V.O.Manturov ∗

, I.M.Nikonov†

October 19, 2022

1 Introduction

Virtual knot theory invented by Kauﬀman [4] in late nineties, has experienced a lot of developments

over the last two decades, among the most important of them, we can mention parity and picture-valued

invariants, [11].

The latter allow one to realise the principle saying that if a diagram is complicated enough then

it realises itself in a sense very close to that one have when working with free groups: the unique

reduced word representing a group element appears in any other element of this group.

In (virtual, free) knot theory, this principle is realised by the parity bracket, [11], an invariant of

knots valued in linear combinations of knot diagrams. For some (odd irreducible) knot diagrams one

has the formula

[K] = K

meaning that the value of the invariant called bracket [·] on a concrete knot diagram Kequals

this diagram with coeﬃcient 1. Taking any other diagram K0equivalent to K, we get [K0] = [K],

which, in turn, mean that the diagram K0“contains” Kas a subdiagram. Such an approach allows

to judge about knots by looking at one particular diagram and estimate various complexities which

open a deeper insight than just numerical or polynomial invariants: we can say much more than just

a bare estimate of crossing number or a certain genus.

In a sense, the parity bracket is a variation of the Kauﬀman bracket where we smooth only even

crossings and evaluate the remaining diagrams (having former odd crossings) to themselves. In the

ﬁrst version of the parity bracket, we are left with ±coeﬃcients, however, many variations of it allow

us to take coeﬃcients in the ring Z[a, a−1] as the classical Kauﬀman bracket, see [2].

Besides that, virtual knots have many invariants valued in free groups and free products of cyclic

groups having similar properties.

All such properties become possible because they use some intrinsic “parity” or “non-trivial ho-

mology” of the ambient space where virtual knots live. Classical knots have no parity and no parity

bracket.

∗Moscow Institute of Physics and Technology

†Moscow State University

arXiv:2210.09689v1 [math.GT] 18 Oct 2022

However, classical knot theory has many instances of non-trivial homology and free groups, in

particular, the braid group conﬁguration spaces and braid group faithful action on free groups.

In 2015, the author introduced a two-parameter family of groups called Gk

nand formulated the

following principle [5].

If dynamical system describing a motion of nparticles possess a nice codimension 1property

governed exactly by kparticles then such dynamics possess invariants valued in groups Gk

n. Without

going into details of the groups Gk

n, we mention just one property for dynamics of motion of npoints

on the plane: for k= 3 we can take the property “three points are collinear”.

In some sense, the invention of the groups G3

nwere an attempt to invent a substitute for parity in

the classical setup: the ambient space R2(or R3) does not possess any homology, but when dealing

with braids, the punctured plane does contain homology groups.

Lots of invariants of braids appeared since that time; moreover, the Gk

ntheory works well for

studying fundamental groups of other conﬁguration spaces.

The main obstruction from extending this theory from braids to knots was a very important

condition that the number of particles of dynamical system should stay ﬁxed during the motion; in

particular, a knot in R3with its maxima and minima does not satisfy such conditions.

In [13], we take the above approaches together and convert classical braids into braids in the

cylinder, and then to braid diagrams drawn in surfaces of higher genera.

In the present paper, we address the problem how to get a map from knots in the cylinder and on

the thickened torus to some (generalisation of) virtual knots called virtual-ﬂat knots.

The main feature of this construction is that starting from objects with rather modest homology

group (H1(S1×I1) = Z) we can construct virtual knots of arbitrary high genus hence having lots of

features.

The main construction takes a diagram on a cylinder (torus) and adds some “invisible” crossings

which gives rise to a diagram which can be formally immersed but not embedded (drawn) on the

cylinder (torus) and living comfortably in thickened surfaces of higher genera.

This allows one to “pull back” invariants of virtual theory to the theory of knots in the thickened

cylinder (torus) where the parity bracket and other picture-valued invariants are not strong enough.

This project was initiated in the paper [13]

at the level of braids where a map from classical braids to (a generalisation of) classical braids led

to new representations of the classical braid group.

Aﬂat-virtual link diagram is a four-valent graph on the plane where each vertex is of one of the

following three types:

1. classical (in this case one pair of edges is marked as an overcrossing strand).

2. ﬂat;

3. virtual.

The number of link components of a ﬂat-virtual link diagram is just the number of unicursal

component of such. A ﬂat-virtual knot diagram is a one-component ﬂat-virtual link diagram.

The number of components is obviously invariant under the moves listed below, so, it will be

reasonable to talk about the number of components of a ﬂat-virtual link.

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Mapsfromknotsinthecylindertoat-virtualknotsV.O.Manturov*,I.M.NikonovOctober19,20221IntroductionVirtualknottheoryinventedbyKauman[4]inlatenineties,hasexperiencedalotofdevelopmentsoverthelasttwodecades,amongthemostimportantofthem,wecanmentionparityandpicture-valuedinvariants,[11].Thelatterallowoneto...

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Maps from knots in the cylinder to at-virtual knots V.O.Manturov I.M.Nikonov October 19 2022

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