Topology Of spaces

in dirmens]on

f knots

Ryan Budney

rybuCrmpirm―bonn.mpg.de

Max Pianck lnstitute

BOnn,Germany

Ｏ

　

｛３

Embedding spaces

Definitioni For compact manifOlds ittr and N

let Ernb(M,N)be the set of embeddings of

y in N,Let κ 角=Emb(R,Rり be the Setof embeddings of R in Rtt which agrees with

t h e i n c t u s i o n密一→(何,o,…・,0) O u t s i d e o f I =

[ - 1 , 1 1 .

ExarTBpie: The C'0-uniform topology is the urrong

topology on embedding spaces.Consider」 F:

[ 0 , 1 ] ×κ角→ κt t w h e r e

( 1 - t ) メ住詳考・露) を く( 1

露 t = 1P(サ,メ)(密)=

ｒ

ブ

ヽ

ｋ

を=と

The aright'tOpOlogy on,に 句

Definition: The C!ん ―metric on κ tt is given by

】ん(メ,9)工=竹 文預牝cR{杓 /】Σ侯=01」Dあメ(サ)一 五メθ(サ)12}

The topology on,に 角 is defined to be the one

generated by alithe Cん―metHcsん∈(o,1,2,…・}.

The topology on Emb(7,N)iS dettned analo―

9ousiy,via charts.

VVhy should l care? (■)

`the long―ciOsed relatiOn'

↑:R→RRヽ●

↑:s'―→ぎ十

転

″キヤ一一一

PropOsltiOn:↓

Emb(Sl,S句と学_Sο角+1焦・‖いほ↓ぼ五吐 仏 | ぇ8 9 )

ほもは,「,″

3】 ・ar ‐

二“↓仲′対)

ヤL 悔

“・SCちは3/stt ι

日`fChPしγ

レ

A .

い

8 1 B I ・・

1 ' , 2

VVhy`deform

Proposition:

embeddings of

T A

〈fヽ/とど

だ

　　か

】ヽ

r

s こヽこぅRAt '▼R体lFV屯品「

ご

うｆ

Ｓ

∂叫‖ゴVVC‖↓

f・oL やExalmple: Z生 後 κ4ニ

T"法 ょ一

should l care? (2)

spun knots'(Litheria nd)

津i s A→陥Elements of Tぅ,Ctt produce tspun'

sづ+1,sin Rづ 十角.

L一→孔患Цちても「,貿tA,

Rぢ十れ―

tSVl

う

停 r

VVhy should l care?(3)

syrnrnetry properties

G i v e n a k n o t メC κ3 1 e t κ3 ( メ) d e n O t e t h e p a t h …component of κ3 COntainingメ.

Proposition: An invertible knotメ c κ 3 iSstrongly invertible if and only if any`inversion'

involution r given by 2T― rotation about an axis

perpendicular to the long axis has a fixed point.

k → 氏So the study of syrnrnetry properties becomes

fixed―point theory.

の

，

３＜

び

κ↓の

や

３＜

３

κｒ

Early results

Theorerl: ( V V h i t n e y ' 3 6 ) I f づ≦れ- 4 t h e n

Tづ丈;れ==0

Theorem: (smale'59,Earle― Eelis'67)κ 2iScontractible. 疼ぅ D,rrD4営平

Theoreim: (Gramain'77)Ifメ iS a nOn―trivial

iong knot in R3, 2T rotation about the long

axis generates an infinite―cyclic subgroup of

Tlκ3(メ)・

ヽ

β一ＶＴ

ｒ

２

b″c研『K律'→LD

More recent resu:ts

Theorenl: (HatCher'83)

●The components of κ 3 are【 (T,1)'S.

Propositioni (TurChin'ol)POiSSOn bracket

on g2_page of vassiliev spectral sequence for

r*κ 角.

Theorem:(B'03)There exists an action of

the operad of 2-cubes on,電 3・

●‐The cornponent oftr

tractible.やめ9βPD3,

Theorem: (vaSSiliev'9o, Goodwillie… 付ヽeiss'99)COnstruction of a spectral sequence for

月「*κれ and盟 「*κれ,convergence resuits for SS

W h e n 句> 4 .

pPA ( s i n g i e ) t i t t l e角 …cube i s a n e m b e d d i n gう :

Pち 一)P such that あ = [1× ..・×仇 where ごづ :Iす I h a s t h e f o r r n Jづ(サ) = a , t t t bをW i t h a t > 0 .

正三 [- 1 , 1 ] .

AJ―tu ple(あ1ル2,…・,あJ)isサ littie角―cubes'if

■)あぁiS a littleれ―cube for aH l≦づ≦J.

2)The inteHOr of the images of tt andあたaredittOint providedぢ井 ん.

The space of J littte角―cubes is denoted cれ(ブ).

C句 :〓=てC角(o),Cれ (1),C句 (2),・・・}iS the Operad Of

littie角―cubes,the operad structure being given

by the composition rnaps

C 句( ゴ) ×( Q ( た1 ) ×… X 晩( り) →晩( ん1 + ・+ 均 )

Q(す )×(晩(ん1)× …。X晩 (崎))→ Cれ(んlⅢ …十均)

Example:角 =2,ゴ =2,ん 1=3,た 2=4.

C2(2)× C2(3)× C2(4)。→〉C2(7)

1 2

□

□□

□

回

□

s●ユ ↓

There is an identity element r地角cc角(1),andC句( J ) a d r n i t s a f r e e a c t i O n o f t h e s y m r n e t t t cg r o u pだ,ヶ.

An action of

spaceズ is a

which satisfy

the operad of littie

collection of rnaps

んが C 角( あ) ×X を→ X

three axiorns.

1)Identity.馬1(r亀れ,何)=密 fOr all密∈ズ.

2)Symmetry.馬 づ(五.σ,何.σ)=馬 づ(島,″)fOr allろ∈

晩 (す)and密 ∈ズタ.

れ―cubes on a

3)Associativity.

モn(も)x(て.(よt)xメ _ゥ c句(す)X― ↓

C 句( ん1 + … ・十 り ×評 1 + …十崎_ _ → ズ

牝七・竹

| ズ′

k r e

cornrnutes.

Observationi ttf a spaceヌ i adrnits an action

of C角 ,then T。 ん2:TO(抗 (2)× (TOズ )2→ T。 ズ

gives a monoid structure on Tox,

Theorem: (BOardrnan― vogt'68,May'74)If

ズ adrnits an action of ctt making T。 易f into a

l号 】!c:|:1言 'f°

rne Space X′ ・

atible、何ith the connect…

sum operation on T。 ,(3 WOuld then be a `pull

one knot through the other'farnily of rnaps.

eg: ダ:〓=2 ¬

―S_O=「 輪コ

- 6 _ o

Theorem:(B'03)There iS an actiOn of the

operad of 2-cubes on,(3 COmpatible with the

connect― surn operation.

Definit3oni Given a pointed space y there ex―

ists an otteCt Ca‖ed the afree c角―space on y'

denoted ctty defined to be(□鱗≧。C句(ん)Xずんyん)/付.where the equivalence relation tt is generated

by

((う1,・… ,あヵ…・

((う1ガ…,%…,あた) , ( 7 1 ,・… , * ,…・

, 7ん) )

ルか,(%…浮_の

Observationi

then C2Cて日て*})Let.X'be an

壬 □濯=。C2(ん)

unpointed

×軌 ズんspace

Theorem:(sChubert'49)Let?be the sub―

space of,c3COnSisting of the prime knots then

the cubes action on κ3 reStricts to a rnap

C2(7)と」(*}):=(□鱗≧oC2(ん)×sた?ん)→ κ3which induces a bueCtiOn Toc2(クロ〈*})―→TOκ3・

Pと!1幸r_ギ4Observation: Given メ C'(3, the CornponentOf C2(1″□〈*})COrresponding toメhaS the hOmotopy―type of c2(ん)×Σメ田推=l κ3(洗)Whereメ iS theconnect sum ofメ1,た,…・,れ∈ク,where Σメ⊂Sttis the Young subgroup corresponding to the

r e l a t i o n t t t J ←→κ3 ( 洗) = κ3 ( 乃) ・

T h e o r e m : ( B ' 0 3 ) T h e m a p

C2(?□ 〈*})→ κ3

is a hornotopy一eq uivalence.

Tわ ols for

Theorenl:

tractible,

Corollary: Given a

be the complement

proving theorems about K,3

( H a t C h e r ' 8 3 ) D i f F ( D 3 )●

is con―

◆R→Rけ β :ぎ→ 、3

10早9 knotメ∈サに3 1etOfメin R3,then

Background: An old

theory was, 4when is a hornotopy equivalence

of 3-rnanifolds hornotopic to a difFeornorphisrn7'

This question was essentia‖ y resolved via along chain of work started by peOple like Seifert,

Hopf, Kneser,Miinor and vvhitehead,and end―

ing in the work of vvaldhausen, Haken, 」 aco,

Shalen,」 ohannson, Thurston and MostOw.

The`primary rnachine'is ca‖ ed the」 aco―shalen…

Johannson(」 S」)deCOrnposition of 3-rnanifolds,

This is what we use to study BDifF(qメ,∂qメ).

隼,ぶvβlet砕

Jaco― shalen― Johannson

decomposition of 3-「 manifolds

A canonical decomposition atong spheres and

tori.

Theorem: (Kneser'29,M ilnor'62)Every

cornpact connected orientable 3-rnanifold A〃

is a connected sum of a unique co‖ ection of

p t t m e 3 - m a n i f o l d s y = M l 非晦 井 … 非比 .

s hミ→ 煎.Definition: A torus tr in a 3-rnanifold 几イ isi n c o m p r e s s i b l e i f T l T →T l 肋r i s i t t e C t i V e .

Theore口 1: (」 acO, shalen, 」 ohannson '78)

Every prime 3-rnanifold」 肋「has a co‖ ectiOn挽 ,7め,・…,7抗 Of digttOintly embedded incompress―i b l e t o H s u c h t h a t y l□推1乳にa d i t t O i n t u n i o no f 3 t o r o i d a l a n d s e i f e r t f i t t r e d m a n i f O l d s . M O r e ―

over,ifttis the rninimal such number,then the

tori are unique up to isOtopy.

A的6fユ|千 もよ 洵

や受9β A亀 4ため毛ど も

立 ♪名ruュ

JSJ― decompositions of

knot complements

Definitioni The JS」 ―graph associated to a prirne

3-manifold A√ is the graph whose vertices are

the components of yl□推1乳and whOSe edgesare〈挽,め,…,軌}

Theorem: (Alexander/sChOenflies)Knot com―

plements are prime.

Theorem:(`Generalised Jordan curve The―

Oremり Every ciosed,connected,embedded sur―face ln s3 seperates S3 intO two components.

Coro‖ ary: of a knot comple―

1中L調ゑ浮終

ｒ

智

ａ

ａ　ｓ

ｒｏ　ｎｔ

Ｏ　ｅ

Ｃ　ｍ

ご。AP・

flAせ,前。1″ て♪(↑よ

JSJ― trees of knot complements, asplicing'

Definition:

An(れ +1)―COmponentiinkあ =(あ 0,あ1,…・,あれ)is a KGL(knot― generating…tink)if the sub_

link(あ1,あ2,・・・,あれ)iS the句_component un―

link. 。っ″" θ ″ra可颯=θ

―― ・

A knot orlink is said to be compound ifits

complement has a non― trivial JsJ―decorn p…

osition

Two non― compound KGLs

DefinitiOn: Given a KGLろ =(あ 。,あ1,・… ,あ句)letん=(ん1,ん2,…・,ん角)be dittOint embeddingsO f l―∞,∞]x D 2 i n t h e c o m p l e m e n t g O f t l∪

([―∞,∞]×D2)∩bσ=んぢ(〈0}×D2)a10ngitu_

偽鶴

↑ ん2

dinal disc forあ か

Define an

identity on

e m b e d d i n g

σ―口推1づ碗o(んづ)!.…

to be the

VVe fix tt non―trivial knots p E=(91,92,・ … ,9句)C

κtt and define」R to beんづo gt oん声l onあ陶。(んぁ).

Definition: &〔ふt偲ガ確クタ

ク※う =兄 ぁ(あ0)一

9'Xあ

〕 tri「

SI′ 次 CoAVtAII・ 0へ)

せｒｌ、

βゝ

巳 ):=偉Rxげ→RxげSUPFtrヽ c IXぬ詩1

cn塩』Jh8 u/

~ ― ‐ ― ツ ●

ヽ―

ノ７

Ｊ

官

ｔ態

倉

■

Ｊ

　

　

　

　

　

′

′ｒ

ヽ後

一

い

１ｌＴ佑

‥―

ぬ

ド

ーー「ヨ

り０ｒ

Ｆ

Ｅ

ヽ文

）

βトーー→ 食1,

SIPi・!ざ aキ 食ムrc

∴ヨぃがじsけだ吃→υ

tRX争]

Some results with F.Cohen.

PropositiOni 五 恥oC3;Q)iS a free Poisson aト

gebra.

PropositiOn: T・ he unknot is the only knot

whose component in,c3haS trivial homology.

P r o p o s i t i O n : T・ he u n k n o t i s t h e o n i y k n o t

whose component in Emb(Sl,S3)haS n0 2-

torsion in its homology.

PropositiOn: ダ *(κ3;Z)COntains torsion of引l order立 て 仲

PropositiOn: rl(κ 3;Z)iS a direct sum ofcopies of Z and Zと .

PropositiOni Odd!件 DtOrSion does not exist

in翌托け(κ3;Z)prOVidedづ <2p-2.

Definitioni ェfr C 丈 ,3 iS a nOn―trivial knot,G*(メ)∈耳1(κ3(メ)iZ)denOtes the Gramain cy―cle, the irnage of the Grarnain element under

the Hurewicz map.

PropositiOn: (B'05)If r is non_trivial there

exists a pttmitive cocycle C*(メ)in「1(κ

3(メ);Z)such that

(G*(メ),G*(メ))三角

where r iS a tonnect―surn of tt prime knots,

Theorern: (SChubert'53)Ifメ ∈κ3haS anon―trivial」sJ―decomposition thenメ =g閑 あfor a co‖ection of non― trivial knots g and a

n o n―co r n p o u n d K G Lう 。

Theorerm: A list of the non― compound KGLs:

1. Torus knots.

2. 2-cornponent seifert links with One corn―

ponent unknotted.

4.Hyperbolic KGLs.

亀

A`generic'」 SJ―tree

陥。ヰ食Theorem: (Alexander/Papakyriakopoulous)A torus embedded in S3 bounds a solid torus

on at ieast one side.

も3

The previous theorern foHows fr6rn results of

Burde and Murasugifor(1),(2),(3),and Thurston

for(4).

How spliCing arrises

イをノ

Theore口 1: (SChubert'53)Given tt dittOint

non―trivial knot complements in s3, they are

uniinked.

Exalmples of g図 乃

8合争Z々

ぜ

↓

ノ

ー

‥

′

′

Ｇ宅

the homotopy type of K,3

page■ of 2

Theorem: (HatCher'83)The cOmponent of

the unknot in iに3 iS COntractible.

nみ2Theorem:(B'03)If a knOtメにa COnnected―sum of tt pttme knotsメ1,れ,…・,あthen

κ3(メ)堂CttR2×Σr rl κ3(免)ぢ_ぃ1

Σメ⊂Stt iS defined as the subgroup of Sれthatpreserves the partition of〈1,2,・…,角}definedbyぢ～ブ⇔κ3(免)主κ3(乃)・

the homotopy type

page 2 of 2

Theorerl: (Hatcher 'o2)If a knOt メ isc a b l i n g o f a k n o t θ t h e n κ3 ( メ) 壁S l X κ3 ( g )

，すＪ陥

字

げＳ

封

焔鮪伍ｒ

ｆＯ

ヽ

、

‐

′

／

ヽｌ

ノ●

２９／ｔヽ３κ

句正

〓．●

を

TheorerTB: (B 705)Ifメ

withあ a hyperbolic【 Gあ

主 (9 1 , 9 2 ,…・,9角)図五

then

とこごと̀ ′と,′,"′転

Where Aメ ⊂ Sο 2 iSithe rnaxirnal group ofisom

あ at extend to difFeomorphisrns oi

ゴ呈,acting by transiation onあ。,t h e r e p r e s e n t a t i o n ムメ→ 球 = T O D i f F ( u 推1 あぢ)naturally acts On rI侠=1,c3(9う)by perm utation

ー

of coordinates a

_止 二8ハサ((ti Jヽ

塩ム Zと協

ヽ