Introduction to Arnolds J-Invariant

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Introduction to Arnold’s J+-Invariant

Alexander Mai

Balex.mai@posteo.net

September 2022

University of Augsburg

arXiv:2210.00871v1 [math.DG] 30 Sep 2022

Table of Contents

Contents

Abstract 1

Introduction 1

1 Basics of immersions 3

2 Events during homotopies of immersed loops 8

3 The J+-invariant 10

3.1 Properties of J+..................................... 10

3.2 Calculation of J+..................................... 11

3.2.1 Viro’sformula .................................. 12

3.3 Exercises ......................................... 14

4 Advanced Topic: Interior sum of immersions 16

4.1 Rotation number from winding numbers . . . . . . . . . . . . . . . . . . . . . . . . 17

4.2 Proofofthetheorem ................................... 21

4.3 Corollaries ........................................ 24

Appendix 29

Solutions ......................................... 29

References 39

Introduction

Abstract

We explore Arnold’s

J+

-invariant of immersions – planar smooth closed curves with non-vanishing

derivative, at most double points and only transverse intersections – and computation methods like

Viro’s sum, among others.

Only basic undergraduate mathematics is needed to understand the contents of this introductory

paper and everything we need that is above that is recalled or introduced. Examples, exercises and

solutions are included for practice.

Introduction

In this introductory paper will thoroughly walk through the following: A regular immersed loop is a

smooth map of a circle into the plane, i.e.

q:S1→C,

with non-vanishing derivative. We identify

the map with its image

K=q(S1)⊂C,

ignoring its parametrization and orientation. We call an

immersed loop

generic

if it has only transverse self-intersections and all of them are double points. For

us all loops of interest are generic immersed loops, which we will simply call

immersions

if not explicitly

stated otherwise.

Vladimir Arnold

introduced three invariants for such immersions [Arn93], of which the invariant

J+

is of special interest for this introductory paper. If we consider an immersion during a regular homotopy

with only isolated non-generic moments, then the

J+

-value of the immersion changes only under

the ﬁrst of the three so-called disasters – direct self-tangencies, inverse self-tangencies, triple point

intersections. If there are two immersions of the same rotation number, then by the

Whitney–Graustein

Theorem

one can be obtained from the other through a regular homotopy. Two such immersions have

the same

J+

-value if and only if during a regular homotopy from one to the other, the number of positive

direct self-tangencies is the same as the number of negative direct self-tangencies. As a consequence,

two immersions can only be homotopic to each other without direct self-tangencies – a tangential

double point where the directions agree – if their J+-value is the same.

Homotopies of immersions without direct self-tangencies are of interest for applications in planar

orbital physics – which are not discussed in this paper – where the Hamiltonian is conserved as the

sum of the potential and kinetic energy. For instance if the orbit of a particle moving in a conservative

force ﬁeld changes, direct self-tangencies cannot occur, but inverse self-tangencies can occur as long

as velocity-dependent forces are present, like the Coriolis force or the Lorentz force. Without any

velocity-dependent force neither direct nor inverse self-tangencies would be possible.

Other invariants based on

J+

can yield further applications, as for instance

and

introduced

Cieliebak, Frauenfelder and van Koert

for any immersion

K⊂C∗,C∗:=C\ {0},

see [CFK17]. They

are invariant under Stark–Zeeman homotopies (see [CFK17]), which are of special interest for orbits of

satellites in space but are not explored in this introductory paper.

After we ﬁnish the introduction to

J+,

as an extra, an approach to calculate

J+

of interior sums (a

notion to be introduced in this paper) of immersions is developed. We take any two immersions

K, K0,

put

into a connected component

C\K,

cross-connect the two immersions at two arcs where the

Introduction to Arnold’s J+-Invariant 1

Introduction

orientations match, and call the resulting immersion

K×.

We then show that the following equation

holds:

J+(K×) = J+(K) + J+(K0)−2·ωC(K)·rot(K0),

with ωC(K)the winding number of Karound Cand rot(K0)the rotation number of K0.

In order to prove this, in Chapter 4.1 we ﬁnd and prove an equation for the computation of the

rotation number of an immersion

that only uses the winding numbers of the connected components

C\K

(denote them as

ΓK

) and the index of each double point (denote them as

), which we

deﬁne to be the arithmetic mean of the winding numbers of the four connected components adjacent

to a double point. We obtain the equation

rot(K) = X

C∈ΓK

ωC(K)−X

p∈DK

indp(K).

First we recall some geometry basics in Chapter 1and then introduce a few concepts of changing

curves and events that can occur in Chapter 2.

Acknowledgements: This introductory paper would not exist without the successful completion

of my bachelor thesis, which was only possible thanks to the extraordinarily patient guidance and

motivation from Kai Cieliebak, Urs Frauenfelder, Ingo Blechschmidt, Julius Natrup, Florian Schilberth,

Milan Zerbin, Leonie Nießeler and other friends from the University of Augsburg.

2 Introduction to Arnold’s J+-Invariant

1 Basics of immersions

Deﬁnition 1.1 (Immersed loop) An immersed loop is a regular loop, i.e. a smooth map

q:S1→C

with non-vanishing derivative, which we will, by slight abuse of notation, from here on identify

with its oriented image K=q(S1)⊂C.

This abuse of notation is not dishonest, as we will not need any explicit deﬁnitions for immersed

loops. Instead, we will draw many pictures – in our heads and in this paper – to talk about immersed

loops. Note that orientation preserving reparametrizations of the immersion – i.e. changing how “fast”

the curve is traversed without changing the direction – have no effect on the oriented image and are

consequently ignored from here on.

Remark. In general, a path in a topological space Xis a continuous map f: [0,1] →X. It is called a

closed path or a loop if f(0) = f(1),which is equivalent to deﬁning a loop as a path with S1,the circle, as the

domain of the path. A path (or loop) is called regular if it is smooth and has non-vanishing derivative, so the

derivative is non-zero everywhere. The image of a path (or loop) is often called a curve. This is just a reminder

– all curves in this paper will be regular and closed, so we do not need these general deﬁnitions.

Any curve that we can draw on paper, taking the following rules into consideration, is a valid visual-

ization for an immersed loop:

•ends where it started (loop),

•no interruptions (continuous),

•no edges (smooth, derivative never zero),

•is given one of two possible directions (oriented).

The pictures in Figure 1represent immersed loops.

Figure 1: Examples of immersed loops.

The pictures in Figure 2do not.

Introduction to Arnold’s J+-Invariant 3

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IntroductiontoArnold'sJ+-InvariantAlexanderMaiBalex.mai@posteo.netSeptember2022UniversityofAugsburgTableofContentsContentsAbstract1Introduction11Basicsofimmersions32Eventsduringhomotopiesofimmersedloops83TheJ+-invariant103.1PropertiesofJ+.....................................103.2CalculationofJ+........

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