knots in four dimensions - an introductionlph5s/t/2.pdf · what is a surface knot? first recall the...

Knots in FourDimensions

LawrenceHook

Overview

What is asurface knot?

RosemanMoves

Trivial SurfaceKnots

Roseman Moves

MovingPictures

Spinning

Slice vsRibbon Knots

Slice Knots

Ribbon Knots

Slice-RibbonConjecture

Knots in Four DimensionsAn Introduction

Lawrence Hook

University of Virginia

MATH 4840 - 2016


LawrenceHook

Overview


RosemanMoves


Roseman Moves

MovingPictures

Spinning


Slice Knots

Ribbon Knots


Overview

What is a surface knot?

Roseman Moves

Movies / Motion Pictures

Construction by Spinning

Slice vs Ribbon Knots


LawrenceHook

Overview


RosemanMoves


Roseman Moves

MovingPictures

Spinning


Slice Knots

Ribbon Knots


What is a surface knot?

First recall the definition of a conventional knot in 3-space.

Definition

A knot is a closed 1-manifold embedded in R3 with a singlecomponent (i.e., an embedded circle in R3).

Now, in one dimension higher.

Definition

A surface knot is a 2-manifold embedded in R4.


LawrenceHook

Overview


RosemanMoves


Roseman Moves

MovingPictures

Spinning


Slice Knots

Ribbon Knots


Trivial Surface Knots

Conventional knots (i.e. the image of S1 in R3), are trivialwhen they are equivalent to the unknot.

For surface knots, the definition is a little more complex. Aknot is trivial when it is equivalent to a standard surface.These are the orientable surfaces in R3.

Figure: Trivial surface knots


LawrenceHook

Overview


RosemanMoves


Roseman Moves

MovingPictures

Spinning


Slice Knots

Ribbon Knots


Trivial Surface Knots

To better determine when a surface knot is trivial

Proposition

For an orientable surface link F in R4, the following areequivalent

F is trivial (unknotted)

There is an embedded 3-manifold M in R4 with ∂M = Fsuch that each connected component of M is ahandlebody.

F is equivalent (or ambient isotopic) to a surface linkwhich is contained in a 3-plane in R4.


LawrenceHook

Overview


RosemanMoves


Roseman Moves

MovingPictures

Spinning


Slice Knots

Ribbon Knots


Dennis Roseman

“Reidemeister-Type Moves for Surfaces in Four-DimensionalSpace”

Figure: D. Roseman

Fun fact: enjoys tenor banjo and organic gardening.


LawrenceHook

Overview


RosemanMoves


Roseman Moves

MovingPictures

Spinning


Slice Knots

Ribbon Knots


Roseman Moves

We covered the three Reidemeister moves in class.

In four dimensions, we have the seven Roseman moves.

Theorem

Any two diagrams of the same surface knot can be related by asequence of Roseman moves.

Roseman also introduced a set of moves for five dimensions.There are twelve of these.


LawrenceHook

Overview


RosemanMoves


Roseman Moves

MovingPictures

Spinning


Slice Knots

Ribbon Knots


Roseman Moves

The following few slides will show the Roseman moves.

These pictoral examples may be opaque but I will show somevideos after!

Of note, I have seen different enumerations of these moves. Iam using the one presented in “Braid and Knot Theory inDimesion Four” by Seiichi Kamada.


LawrenceHook

Overview


RosemanMoves


Roseman Moves

MovingPictures

Spinning


Slice Knots

Ribbon Knots


Bubble Moves

A type I Roseman move corresponds to a type I Reidemeistermove followed by its inverse.

Similarly, a type II Roseman move corresponds to a type IIReidemeister move followed by its inverse.


LawrenceHook

Overview


RosemanMoves


Roseman Moves

MovingPictures

Spinning


Slice Knots

Ribbon Knots


Saddle Moves

A type III move corresponds a type I Reidemeister move.

A type IV move corresponds a type II Reidemeister move.


LawrenceHook

Overview


RosemanMoves


Roseman Moves

MovingPictures

Spinning


Slice Knots

Ribbon Knots


Moves 5-6

A type V move corresponds to a type III Reidemeister move.

A type VI move, also called a tetrahedral move.


LawrenceHook

Overview


RosemanMoves


Roseman Moves

MovingPictures

Spinning


Slice Knots

Ribbon Knots


Move 7

A type VII move is shown below.


LawrenceHook

Overview


RosemanMoves


Roseman Moves

MovingPictures

Spinning


Slice Knots

Ribbon Knots


Videos.


LawrenceHook

Overview


RosemanMoves


Roseman Moves

MovingPictures

Spinning


Slice Knots

Ribbon Knots


Moving Pictures

A moving picture (or movie or tomography) is a series oflink diagrams which are obtained by slicing a surface linkdiagram by parallel hyperplanes.

We saw examples of movies in the presentation of Rosemanmoves.


LawrenceHook

Overview


RosemanMoves


Roseman Moves

MovingPictures

Spinning


Slice Knots

Ribbon Knots


Moving Pictures

A movie is a progression of frames, each of which represents aconventional knot in 3-space. The progression of these framespresents the fourth dimension of a surface knot as time, andthe difference between one frame and the next can be almostcompletely described with ambient isotopy and the threeReidemeister moves. Additional complexity arises from socalled critical points.


LawrenceHook

Overview


RosemanMoves


Roseman Moves

MovingPictures

Spinning


Slice Knots

Ribbon Knots


Elementary Critical Points

1 Maximal points - A trivial loop disappears

2 Minimal points - A trivial loop appears

3 Saddle points


LawrenceHook

Overview


RosemanMoves


Roseman Moves

MovingPictures

Spinning


Slice Knots

Ribbon Knots


non-Elementary Critical Points

There are also non-elementary critical points. Luckily, in thesame way that conventional knots can be deformed so thatcrossings are only between two strands, all surface knots can bedeformed such that their moving picture representation haveonly elementary critical points.

Figure: A non-elementary critical point


LawrenceHook

Overview


RosemanMoves


Roseman Moves

MovingPictures

Spinning


Slice Knots

Ribbon Knots


non-Elementary Critical Points

Figure: Resolving a non-elementary critical point


LawrenceHook

Overview


RosemanMoves


Roseman Moves

MovingPictures

Spinning


Slice Knots

Ribbon Knots


An example moving picture

Figure: Moving picture representation of a spun trefoil


LawrenceHook

Overview


RosemanMoves


Roseman Moves

MovingPictures

Spinning


Slice Knots

Ribbon Knots


Spinning

In 1926, Emil Artin published a paper presenting a way toconstruct surface knots from conventional knots.


LawrenceHook

Overview


RosemanMoves


Roseman Moves

MovingPictures

Spinning


Slice Knots

Ribbon Knots


Low dimension example

The spinning mentioned in this construction is very similar tothe spinning we are familiar with.

Figure: An example of “spinning”


LawrenceHook

Overview


RosemanMoves


Roseman Moves

MovingPictures

Spinning


Slice Knots

Ribbon Knots


Spinning

First Step:Given a knotted circle in R3, first remove a small trivialneighborhood to obtain a knotted arc.


LawrenceHook

Overview


RosemanMoves


Roseman Moves

MovingPictures

Spinning


Slice Knots

Ribbon Knots


Spinning

Second Step:Embed the resulting knotted arc into R3

+ with the twoendpoints fixed in the xy-plane.


LawrenceHook

Overview


RosemanMoves


Roseman Moves

MovingPictures

Spinning


Slice Knots

Ribbon Knots


Spinning

Third Step:This final step involves spinning R3

+ along an axis of2-dimensions which is not very intuitive. However, sinceconventional knots are normally projected on 2-dimensions, wecan instead spin this 2-dimensional diagram in R2

+ around thex-axis.


LawrenceHook

Overview


RosemanMoves


Roseman Moves

MovingPictures

Spinning


Slice Knots

Ribbon Knots


Twist Spinning

Building on the spinning construction, Erik ChristopherZeeman, in 1965 published “Twisting Spun Knots”. Here it isillustrated in a lower dimension. The same concept is used inthe twist-spinning construction for knots in four dimensions(and beyond).

Figure: A twist-spun 0-dimensional knot


LawrenceHook

Overview


RosemanMoves


Roseman Moves

MovingPictures

Spinning


Slice Knots

Ribbon Knots


Slice Knots

Definition

A knot K in R3 is a slice knot if there is a properly embedded2-disk D in R3[0,∞) with ∂D = K [0].


LawrenceHook

Overview


RosemanMoves


Roseman Moves

MovingPictures

Spinning


Slice Knots

Ribbon Knots


Ribbon Knots

Definition

A knot K in R3 is a ribbon knot if it bounds a singular disk Min R3 such that all singularities are ribbon singularities.


LawrenceHook

Overview


RosemanMoves


Roseman Moves

MovingPictures

Spinning


Slice Knots

Ribbon Knots


Slice-Ribbon Conjecture

It is known that every ribbon knot is slice.The converse remains unknown.

Conjecture

A slice knot is a ribbon knot


LawrenceHook

Appendix

For FurtherReading

For Further Reading I

S. KamadaBraid and Knot Theory in Dimension Four2002Fun fact: This monograph has 996 references!

G. FriedmanKnot SpinningOctober 28, 2004

E. DalvitMovies of Roseman moveshttp://katlas.math.toronto.edu/ester/movies/

knots in four dimensions - an introductionlph5s/t/2.pdf · what is a surface knot? first recall the...

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