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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