On knots in overtwisted contact structures

0.32 MB·English

by  John B. Etnyre

ON KNOTS IN OVERTWISTED CONTACT STRUCTURES
JOHNB.ETNYRE
ABSTRACT. Weprovethateachovertwistedcontactstructurehasknottypesthatarerepresentedby
infinitelymanydistincttransverseknotsallwiththesameself-linkingnumber. Insomecases,wecan
evenclassifyallsuchknots.WealsoshowsimilarresultsforLegendrianknotsandprovea“folk”result
2
concerningloosetransverseandLegendrianknots(thatisknotswithovertwistedcomplements)which
1
saysthatsuchknotsaredeterminedbytheirclassicalinvariants(uptocontactomorphism).Finallywe
0
discusshowtheseresultspartiallyfillinourunderstandingofthe“geography”and“botany”problems
2
forLegendrianknotsinovertwistedcontactstructures,aswellasmanyopenquestionsregardingthese

n problems.
a
J

3

1. INTRODUCTION
]
G
SinceEliashberg’sformativepaper[10]classifyingovertwistedcontactstructureson3–manifolds,
S thestudyofandinterestinsuchstructureshasbeenminimal. However,inrecentyearstheyhave
.
h beentakingamorecentralroleduetotheirmanyinterestingapplications—suchas,theconstruc-
t tionofachiralLefschetzfibrations[15]andnearsymplecticstructures[19]oncertain4–manifolds
a
m andtheunderstandingoftheexistenceofEngelstructureson4–manifolds[30]—aswellasthein-
terestingknottheorytheysupport. ThispaperisaimedatstudyingtheLegendrianandtransverse
[
knottheoryofovertwistedcontactstructures. Webeginwithabriefhistoryofthesubject.

2 A Legendrian or transverse knot in an overtwisted contact structure ξ on a 3–manifold M is
v
called loose if the contact structure restrictedto its complement is also overtwisted, otherwise the
5
knot is called non-loose. Though apparentlyknown to a few experts, the first explicit example of
4
7 a non-loose knot was given by Dymara in [9], where a single non-loose Legendrian unknot was
3 constructed in a certain overtwisted structure on S3. More recently, Eliashberg and Fraser [11]
. gave a coarse classification of Legendrian unknots in overtwisted contact structures on S3, see
2
1 Theorem 2.2 below. (We say knots are coarsely classified if they are classified up to co-orientation
0 preservingcontactomorphism, smoothly isotopic tothe identity. We reservetheword classified to
1 refertotheclassificationuptoLegendrianisotopy, andsimilarlyfortransverseknots.) Animme-
:
v diate corollary of this work is that there are no non-loose transverse unknots in any overtwisted
i contactstructure.
X
In [14] it was shown that there are knot types and overtwisted contact structures for which
r
a therewerearbitrarilymanydistinctnon-looseLegendrianknotsrealizingthatknottypewithfixed
Thurston-Bennequininvariantandrotationnumber. Whileitiseasytoconstructnon-loosetrans-
verse knots in any overtwisted contact structure (one just observes, cf. [12], thatthe complement
ofthebindingofasupportingopenbookdecompositionistight),twonon-loosetransverseknots
withthe sameself-linkingnumberswerefirstproducedbyLisca,Ozsva´th, StipsiczandSzabo´,in
[27],usingHeegaard-FloerinvariantsofLegendrianandtransverseknots.
TherehavebeenveryfewresultsconcerningtheclassificationofLegendrianortransverseknots
inovertwisted contactstructure(asopposed tothecoarseclassification),buttherehasbeensome
workgiving necessaryconditions forthe existenceof aLegendrianisotopy, seeforexample[6, 8,
11].
Leaving the history of the subject for now we begin by recalling a version of the Bennequin
boundfornon-looseknots. Thisresultfirstappearedin[8]whereitwasattributedtoS´wia¸tkowski.
Proposition1.1 (S´wia¸tkowski, see [8]). Let (M,ξ) be an overtwistedcontact 3–manifoldand L a non-
looseLegendrianknotinξ. Then
−|tb(L)|+|r(L)|≤−χ(Σ)
foranySeifertsurfaceΣforL.
1
2 JOHNB.ETNYRE
We sketch a simple proof of this result below. We now observe a relation between non-loose
transverseknotsandtheirLegendrianapproximationsaswellasnon-looseLegendrianknotsand
theirtransversepush-offs.
Proposition1.2. IfT isanon-loosetransverseknotthenanyLegendrianapproximationofT isnon-loose.
IfLisanon-looseLegendrianknotthenthetransversepush-offofLmayormaynotbenon-loose.
TheprevioustworesultsimplythataversionoftheBennequinboundfortransverseknotsina
tight contactstructure alsoholds for non-loose transverseknots. This isin starkcontrastto what
happensforLegendrianknotsasProposition1.1indicatesandEliashbergandFraser’scoarseclas-
sificationofunknots,giveninTheorem2.2below,confirms.
Proposition1.3. Let(M,ξ)beacontact3–manifoldandK atransverseknotinξwithSeifertsurfaceΣ. If
sl(K)>−χ(Σ)
then K is loose (and, of course, ξ is overtwisted). In particular any non-loose knot K in an overtwisted
contactstructuresatisfiestheBennequininequality
sl(K)≤−χ(Σ).
1.1. ThecoarseclassificationoflooseLegendrianandtransverseknots. Thefollowingtwotheo-
remsmakeprecisethewell-known“folk”theoremsthatloose Legendrianortransverseknotsare
coarselyclassifiedbytheirclassicalinvariants.
Theorem1.4. Let (M,ξ)beanovertwistedcontactmanifold. Foreachnull-homologousknottypeKand
eachpairofintegers(t,r)satisfyingt+risodd,thereisaunique,uptocontactomorphism,looseLegendrian
knotLintheknottypeKwithtb(L)=tandr(L)=r.
Recall that for any Legendrian knots L we must have tb(L) + r(L) odd, so the above theorem
saysanypossiblepairofintegersisrealizedbyauniquelooseLegendrianknotinanyovertwisted
contactstructure. Fortransverseknotswehavethefollowingresult.
Theorem1.5. Let (M,ξ)beanovertwistedcontactmanifold. Foreachnull-homologousknottypeKand
eachoddintegersthereisaunique,uptocontactomorphism,loosetransverseknotT intheknottypeKwith
sl(T)=s.
Againrecallthattheself-linkingnumberofanytransverseknotmustbeoddandthusthetheorem
saysthatanypossibleintegerisrealizedbyauniqueloosetransverseknotinanovertwistedcontact
structure.
ThesetwotheoremsfollowdirectlyfromEliashberg’sclassificationofovertwistedcontactstruc-
turesandacarefulanalysisofhomotopyclassesofplanefieldsonmanifoldswithboundary,which
wegiveinSection4. Theorem1.4alsoappearsin[11]thoughthedetailsofthe homotopy theory
werenotdiscussed,andwhilethesedetailsarefairlystraightforwardtheydonotseemobviousto
theauthor. Inparticular,whenstudyingthehomotopyclassesofplanefieldsthereisbotha2and
3-dimensionalobstruction tobeingabletohomotope one planefield toanother. The factthatthe
3-dimensional obstruct is determined by the Thurston-Bennequin invariant and rotation number
seems, atfirst, a little surprising until one carefullycomparesthe Pontryagin–Thom construction
with a relativeversion of the Pontryagin–Thom construction. Geigesand independently, Klukas,
in private communication, informed the author of another way to deal the the homotopy issues
and prove the above theorems. Similar theorems, with extra hypotheses, concerning Legendrian
isotopywereprovenin[6,8].
In [28] Chernov defined relative versions of the rotation number for all (not necessarily null-
homologous)Legendrianknotsaslongastheambientmanifoldisirreducibleandatoroidalorthe
Eulerclassofthecontactstructureisatorsionclass(orthecontactstructureistight).In[1,28]hede-
finedrelativeversionsoftheselflinkinginvariant(nottobeconfusedwiththeself-liningnumber
ofatransverseknot)forall(notnecessarilynull-homologous)framedknotsinatoroidalandother
manifolds. This gives the definition of the relative Thurston-Bennequin number for Legendrian
knotsincontactmanifoldsofsuchtopologicaltype. Thereshouldbetheoremsanalogoustothose
abovefornon-null-homologousknotsinthissituation,buttheprecisestatementswouldnecessar-
ilybemorecomplicated. Itwouldbeinterestingtoseethepreciseextensionoftheabovetheorems
ONKNOTSINOVERTWISTEDCONTACTSTRUCTURES 3
toallnon-null-homologousknotsinallovertwistedcontactmanifolds. See[6,8]forpartialresults
alongtheselines.
1.2. Non-loosetransverseknotsandthecoarseclassificationoftransversefiberedknotsrealizing
theBennequinbound. In[5],Colin,GirouxandHondaprovedthatifonefixesaknottypeinan
atoriodal3–manifoldandatightcontactstructureonthemanifoldthenthereareonlyfinitelymany
LegendrianknotswithgivenThurston-Bennequininvariantandrotationnumber. Whilethisdoes
notimplythesamefinitenessresultfortransverseknots,itseemslikelythatsuchafinitenessresult
is true for such knots too. Surprisingly this finiteness result is far from true in an overtwisted
contact structure. We begin with some notation and terminology. We call the open book for S3
withbindingtheunknotthetrivialopenbookdecompositionandsayanopenbookdecompositionis
non-trivialifitisnotdiffeomorphictothetrivialopenbookdecomposition.
If (M,ξ) is a contact 3–manifold and K is a topological knot type we denote by T(K) the set
ofalltransverseknotsintheknottypeKuptocontactomorphism (co-orientationpreservingand
smoothly isotopic to the identity). (We note thatinsome contexts, one mightwantthis todenote
the transverse knots up to transverse isotopy, but in this paper we will only consider the coarse
classificationofknots.) Ifnisanintegerthen
T (K)={T ∈T(K): sl(T)=n}
n
isthesetoftransverseknotsintheknottypeKwithself-linkingnumbern.
Given a null-homologous knot K we will denote the maximalEuler characteristic for a Seifert
surfaceforK byχ(K).
Theorem 1.6. Let (B,π) be a non-trivialopenbook decompositionwith connected binding of a closed 3–
manifoldM andletξ bethecontactstructureitsupports. Denotebyξthecontactstructureobtainedfrom
B
ξ by a full Lutz twist along B. Then T (B) contains infinitely many distinct non-loose transverse
B χ(B)
−
knotsuptocontactomorphism(andhenceisotopytoo).
Thistheoremgivesthefirstknownexampleofaknottypeandcontactstructurewhichsupports
aninfinitenumberoftransverseknotswiththesameself-linkingnumber. Wehaveasimilarresult
usinghalf-Lutztwists.
Theorem 1.7. Let (B,π) be a non-trivialopenbook decompositionwith connected binding of a closed 3–
manifoldM andletξ bethecontactstructureitsupports. Denotebyξthecontactstructureobtainedfrom
B
ξ byahalfLutztwistalongB. ThenT (B)containsinfinitelymanydistinctnon-loosetransverseknots
B χ(B)
uptocontactomorphism(andhenceisotopytoo).
Finallywehavethefollowinglessspecificbutmoregeneraltheoremalongtheselines.
Theorem1.8. LetKbeanull-homologousknottypeinaclosedirreducible3-manifoldM withSeifertgenus
g > 0. ThereisanovertwistedcontactstructureforwhichT (K)isinfiniteandadistinctovertwisted
2g 1
−
contactstructureforwhichT (K)isinfinite.
2g+1
−
Remark 1.9. We note that as mentioned above, and proven in Corollary 2.3 below, any trans-
verse unknot in an overtwisted contact manifold is loose. Hence the previous theorem implies
that the unknot is the unique null-homologous knot in any (irreducible) manifold that does not
have non-loose transverse representativesin some overtwisted contactmanifold (in factall other
null-homologousknotshavenon-loosetransverserepresentativesinatleasttwoovertwistedcon-
tactmanifolds).Thereasonforthisissimplythattheunknotistheuniqueknotwhosecomplement
hascompressibleboundary.
Since, as noted above, Legendrianapproximations of non-loose transverse knots arealso non-
loose, we see that all null-homologous knots have non-loose Legendrian representativesin some
overtwistedcontactstructure. (cid:3)
Usingresultsfrom[18] we canrefineTheorem1.6for hyperbolic knots togive the coarse clas-
sification of transverse knots in a fibered hyperbolic knot type that realize the upper bound in
Proposition1.3.
4 JOHNB.ETNYRE
Theorem1.10. Let(B,π)beanopenbookdecompositionwithconnectedbindingofaclosed3–manifoldM
andletξ bethecontactstructureitsupports. Denotebyξ thecontactstructureobtainedfromξ byafull
B B
LutztwistalongB. IfB isahyperbolicknotthen
T (B)={K }∪{K } ,
χ(B) i i A
− ∗ ∈
whereA=Nifξ istightandA=N∪{0}ifnot;andifξ isanyovertwistedcontactstructurenotisotopic
B ′
toξthen
T (B)={K }.
χ(B)
− ∗
Moreover
• K is loose and the knot K is non-loose and has Giroux torsion i along a torus parallel to the
i
∗
boundaryofaneighborhoodofK .
i
• TheHeegaard-FloerinvariantsofallknotsinT (B)vanishexceptforK ,whichonlyexistsif
χ(B) 0
−
ξ isovertwisted.
B
• AllknotsinT (B),exceptpossiblyK ,ifitexists,becomelooseafterasinglestabilization.
χ(B) 0
−
We note that K in this theorem might or might not be loose after a single stabilization, but it
0
willcertainlybecomelooseafterasufficientlylargenumberofstabilizations.
Remark 1.11. We note that while Theorem 1.7 does allow us to conclude that for the binding of
anopenbook thereisatleastone overtwisted contactstructuresuch thatT (B) isinfinite, the
χ(B)
technologyin[18]doesnotappearstrongenoughtoprovearesultssimilartoTheorem1.10inthis
case. (cid:3)
1.3. Non-looseLegendrianknotsandthecoarseclassificationofLegendrianknots. Wecanuse
thecoarseclassificationoftransverseknotstounderstandsomeLegendrianknotsinthesameknot
types.If(M,ξ)isacontact3–manifoldandKisatopologicalknottype,thenwedenotebyL(K)the
setofallLegendrianknotsintheknottypeKuptocontactomorphism(co-orientationpreserving
andsmoothlyisotopictotheidentity). Ifmandnaretwointegersthen
L (K)={L∈L(K):r(L)=mand tb(L)=n}
m,n
isthesetofLegendrianknotsintheknottypeKwithrotationnumbermandThurston-Bennequin
invariantequalton.
Theorem1.12. Let(B,π)beanopenbookdecompositionwithconnectedbindingofaclosed3–manifoldM
andletξ bethecontactstructureitsupports. Denotebyξ thecontactstructureobtainedfromξ byafull
B B
LutztwistalongB. IfB isahyperbolicknotthenthereisanm∈Z∪{∞}dependingonlyonB suchthat
foreachfixedintegernwehave
L (B)={L }∪{L } ,
χ(B)+n,n ∗ n,i i∈An
whereA =Nifξ istightorn>mandA =N∪{0}ifnot;andifξ isanyovertwistedcontactstructure
n B n ′
notisotopictoξthen
L (B)={L }.
χ(B)+n,n
∗
SeeFigure1. Moreover
• L is loose and the knot L is non-loose and has Giroux torsion i along a torus parallel to the
n,i
∗
boundaryofaneighborhoodofL .
n,i
• TheHeegaard-FloerinvariantsofallknotsinL (B)vanishexceptforL , whichisnon-
χ(B)+n,n n,0
zero. (RecallL onlyexistsifξ isovertwistedandn≤m.)
n,0 B
• All knots in L (B), except possibly L , if it exists, become loose after a single positive
χ(B)+n,n n,0
stabilization.
• ThenegativestabilizationofL isL .
n,i n 1,i
−
As we noted in the transverse case, this theorem provides the first examples of a knot type
andcontactstructurethatsupportaninfinitenumberofLegendrianknotswiththesameclassical
invariants. WealsoremarkthatifthehypothesisonB beingahyperbolicknotisdroppedthen,as
inTheorem1.6fortransverseknots,onecanstillconcludethatthereareinfinitelymanyLegendrian
knotswithfixedinvariantsasinthetheorem,butwecannotnecessarilygiveaclassificationofthese
Legendrianknots.
ONKNOTSINOVERTWISTEDCONTACTSTRUCTURES 5
tb
A
n
A
n
A
n
A
n
r
A
n
A
n
A
n
FIGURE 1. The knots discussed in Theorem 1.12. A dot at (m,n) indicates there
is a non-loose knot with rotation number m and Thurston-Bennequin invariant
n. The subscript on the dot indicatesthe setthatindexes the familyof non-loose
Legendrian knots with invariants given by the coordinates. The line is given by
−m+n=−χ(B).
Wenoticenowthatif−B andB aresmoothlyisotopicthenwecanextendtheaboveclassifica-
tion.
Theorem1.13. GiventhehypothesisandnotationofTheorem1.12,supposethat−B issmoothlyisotopic
toB,where−BdenotesB withitsorientationreversed. Thenforeachfixedintegernwehave
L (B)={L }∪{L } ,
−χ(B)−n,n ∗ n,i i∈An
whereA =Nifξ istightorn>mandA =N∪{0}ifnot;andifξ isanyovertwistedcontactstructure
n B n ′
notisotopictoξthen
L (B)={L }.
χ(B) n,n
− − ∗
ReflectingFigure1aboutther=0linedepictstheknotsdescribedhere. Moreover
• L is loose and the knot L is non-loose and has Giroux torsion i along a torus parallel to the
n,i
∗
boundaryofaneighborhoodofL .
n,i
• TheHeegaard-FloerinvariantsofallknotsinL (B)vanishexceptforL ,whichisnon-
χ(B) n,n n,0
− −
zero. (RecallL onlyexistsifξ isovertwistedandn≤m.)
n,0 B
• All knots in L (B), except possibly L if it exists, become loose after a single negative
χ(B) n,n n,0
− −
stabilization.
• ThepositivestabilizationofL isL .
n,i n+1,i
Remark1.14. ThereisanamusingcorollaryofTheorems1.12and1.13. Consider the setL (B),
r,t
wherethetildeindicatesthatweareconsideringLegendrianknotsuptoisotopynotjustcontacto-
e
morphism(co-orientationpreservingandisotopictotheidentity).Thereisanaturalsurjectivemap
L (B)→L (B). Whileitissuspectedthatthismapisnotinjectiveingeneral(thoughsometimes
r,t r,t
itis,forexamplewhentheambientcontactmanifoldisS3withitsstandardcontactstructure),itis
e
difficultto come byspecific non-injective examples. Inthe following paragraphwe show how to
useTheorems1.12and1.13toconstructthefirstsuchexamples;thatis,exampleswhereoneknows
thereisadifferencebetweentheclassificationLegendrianknotsuptoisotopyandtheclassification
uptocontactomorphism(isotopictotheidentity). Wealsonotethatthisfactprovesthatthespace
of contact structures on M has a non-trivial loop based at the contact structure considered in the
theorem.
6 JOHNB.ETNYRE
Under the hypothesis of Theorem 1.13, given the non-loose Legendrian L ∈ L (B)
χ,i 0, χ(B)
− −
fromTheorem1.12andthenon-looseknotL ∈L (B)fromTheorem1.13weknowL
′ χ,i 0, χ(B) χ,i
and L are contactomorphic by Theorem−1.12. But−notice that they cannot be Legendrian−iso-
′ χ,i
topics−inceapositivestabilizationofL willresultinalooseknotwhileapositivestabilization
χ,i
−
of L will yield a non-loose knot. (Recall stabilization is a well-defined operation and hence
′ χ,i
Lege−ndrian isotopic knots cannot have different stabilizations.) Notice that this fact concerning
stabilizationsdoesnotprohibitL andL frombeingcontactomorphic sinceacontactomor-
χ,i ′ χ,i
phismcanreversethesenseofsta−bilization.−(Toclarifythislaststatement,wenoticefromtheproof
ofTheorem1.13thatL =−L . NowletN beastandardneighborhoodofaLegendrianknot
′ χ,i χ,i
LandletN ⊂N beas−tandardn−eighborhoodofastabilizationofL.TheregionN\N =T2×[0,1]
′ ′
isabasicslice,see[24]moreonthisterminology,andthesignofthebasicsliceisdeterminedbythe
signofthestabilization.Moreover,thesignofthestabilizationisalsodeterminedbytheorientation
onL.ThuswhenconsideringacontactomorphismfromL to−L =L wegetaninduced
χ,i χ,i ′ χ,i
contactomorphism of their complements. Stabilizingone−will add−abasic s−lice to itscomplement
and since the contactomorphism under consideration reverses the orientation on the underlying
knotitwillreversethesignofthebasicsliceandhencethesenseofthestabilization. Wenoticethat
thecontactomorphismofthecomplementsofL andL extendstoaco-orientationreversion
′ χ,i χ,i
contactomorphismofthecomplementsoftheir−stabilizati−ons. Wealsonoticethatthisphenomena
canonlyhappensince,asplanefields,ξand−ξareisotopicandtherotationnumbersoftheknots
inquestionarezero.) (cid:3)
RecallthattwoLegendrianknotswiththesameclassicalinvariantswilleventuallybecomeiso-
topicaftersufficientlymanypositiveandnegativestabilizations.Tryingtounderstandexactlyhow
manyisaninterestingquestion. Wehavethefollowingpartialresults.
Proposition1.15. Let(B,π)beanopenbookdecompositionwithconnectedbindingofaclosed3–manifold
M and let ξ be the contact structure it supports. Assume that ξ is tight and −B is not isotopic to B.
B B
(Here−B denotesB withthereversedorientation.) Considerthecontactstructureξ obtainedfromξ bya
B
fullLutztwistalongB. IfaknotL∈L(B)satisfies
tb(L)−r(L)>−χ(B)
thenitbecomeslooseafter 1(−χ(B)−tb(L))orfewerpositivestabilizations.
2
There is a similar statement for the case when B is isotopic to −B. In the next subsection we
discussthe“geographyproblem”forLegendrianknotsandinparticularattheendofthesection
wediscussthesignificanceofthisproposition.
Remark 1.16. We notice that examining the proof of Propostion 1.15 allows one to drop the hy-
pothesisthat−B isnotisotopictoB iftb(L)+r(L)6=−χ(B). (cid:3)
1.4. The geographyfor non-looseknots. LetK be aknot type on a contactmanifold (M,ξ). We
havethemap
Φ:L(K)→Z×Z:L7→(r(L),tb(L)).
DeterminingwhichpairsofintegersareintheimageofΦiscalledtheLegendriangeographyproblem.
The image if Φ is frequently called the Legendrian mountain range of K because in the case that ξ
is the tight contact structure on S3 the image resembles the silhouette of a mountain range. This
structurecomesfromthefactsthatwhenξisthetightcontactstructureonS3weknowthat(1)the
imageissymmetric,(2)theThurston-Bennequininvariantisboundedabove,and(3)positiveand
negativestabilizationsshowthatthepairs(m−k,n+l),wherek ≥0,|l|≤k,andl+kiseven,are
intheimageif(m,n)is. (RecallthatforanyLegendrianknotLwemusthavethattb(L)+r(L)is
odd,so(3)saysallpossiblepairsofpointsintheconewithwithsidesofslope±1andtopvertex
(m,n) are realized by Legendrian knots that are stabilizations of the given knot.) For a general
contactstructureweonlyhave(3)andifthestructureistight(2).
OnceoneunderstandtheLegendriangeographyproblemforaknottypeKthentheclassifica-
tionofLegendrianknotswillbecompletewhenoneunderstandsthepreimageofeachpairofin-
tegers. DeterminingΦ 1(m,n)isknownasthebotanyproblemforK. Theweakbotanyproblemasked
−
ONKNOTSINOVERTWISTEDCONTACTSTRUCTURES 7
todetermineifΦ 1(m,n)isempty,finiteorinfiniteforeachpair(m,n). AsmentionedaboveifK
−
isahyperbolicknotandξistightthenthepreimageofany(m,n)iseitheremptyorfinite.
If ξ is overtwisted then everypair of integers (m,n)for which m+n is odd is in the image of
Φ. Thusthegeneralgeographyproblemisnotinterestingforovertwistedcontactstructuresandso
weneedtorestrictourattentiontonon-looseknots. LetLnl(K)bethesetofnon-looseLegendrian
knotsintheknottypeK. DeterminingΦ(Lnl(K))willbecalledtheLegendriangeographyproblem
whenξisovertwisted,whichiswhatweconsiderfromthispointon. Similarlywhenwediscussthe
weakbotanyproblemforovertwistedcontactmanifoldswewillonlybeconsideringthepreimage
of(m,n)thatlieinΦ(Lnl(K))(sincebyTheorem1.4weknowthepreimagecontainsexactly1loose
Legendrianknotifn+misoddandnoneifitiseven).
We now summarize what we know about the geography problem for non-loose knots. First,
Proposition1.1impliesthat(assumingtheknotunderconsiderationisnottheunknot!) theimage
of Φ (when restricted to non-loose knots) is contained in the region shown in Figure 2. The four
blacklinesl ,...,l ,arethelines
1 4
±tb(L)±r(L)=−χ(Σ)
(where allcombinations of ± areconsidered). The region is brokeninto 7 subregions, R ,...,R
1 7
asindicatedinthefigure,bythefourlines. TheregionsR areopenregions,whendiscussingthe
i
correspondingclosedregionswewillofcourseusthenotationsR .
i
tb
l l
3 1
l l
4 R 2
1
R R
4 5
R
3
r
R R
6 7
R
2
FIGURE 2. A non-loose knot mustmap tothe greyregionunder the mapΦ. The
blacklines, labeledl ,...,l , breakthe greyregioninto7subregionsasindicated
1 4
inthefigure. TheregionsR areopenregions(thatisnotincludingthepointson
i
theblacklines).
Question1. Cantherebeanon-looseknotwithimageinR orR ?
1 2
Whilewebelievetheanswertothisquestionislikely“yes”,itseemsfairlylikelythattheanswer
tothenext,related,questionsis“no”.
Question2. Cantherebea(m,n)∈R ∪R withΦ 1((m,n))infinite?
1 2 −
8 JOHNB.ETNYRE
WeknowfromColin,GirouxandHonda[5]thatifyoufixatightcontactstructureonamanifold,
a knot type and two integers, there are finitely many Legendrian knots in the given knot type
realizingtheintegersastheirThurston-Bennequininvariantandrotationnumber. Thisseemsclose
toprovingtheanswertothefollowingquestionis“yes”.
Question3. GivenaknottypeKinamanifoldM arethereonlyfinitelymanyovertwistedcontactstruc-
turesonM suchthatKcanhavenon-looserepresentatives?
Infacttheanswertothisquestionwouldbe“yes”ifthefollowinggeneralquestionaboutLeg-
endrianknotscouldbeansweredintheaffirmative.
Question4. GivenaknottypeKinM isthereanumbernsuchthatifLisanyLegendrianrepresentative
inanycontactstructureonM withtb(L)<nthenLdestabilizes?
Of course this question is also very interesting if one first fixes the contact structure and then
asksfortheintegern. IfthismorerestrictedquestionhadapositiveanswerthentheColin-Giroux-
Hondaresultwouldimplythatforanyfixedtightcontactstructureξ onamanifoldM theLegen-
drianknotsinagivenknottypeKwouldbe“finitelygenerated”,bywhichwemeantherewould
beafinite numberofnon-destabilizableLegendrianknots inL(K)suchthatallother elementsin
L(K)wouldbestabilizationsofthese. Thiswouldalsobetrueinovertwistedcontactstructuresup
toLutztwistingalongtheknot.
WethinkitisverylikelytheanswertothefollowingweakerversionofQuestion3is“yes”.
Question5. GivenaknottypeK inamanifoldM arethereonlyfinitelymanyovertwistedcontactstruc-
turesonM suchthatΦcanhaveinfinitepreimageatsomepoint?(Maybeevenatmosttwosuchstructures.)
WiththenotationestablishedforFigure2wediscussProposition1.15. Thepropositionsaysthat
ifLisaLegendrianrepresentativewithinvariantsintheregionR ∪R ,thenonceitispositively
1 4
stabilizedintoregionR orR itwillbeloose. Thisissurprisinggiventhataccordingtothebounds
5 3
giveninProposition1.1aknotwithinvariantsinR couldtheoreticallybestabilizedpositivelyan
4
arbitrarynumberoftimesandstaynon-loose.
Acknowledgments: Theauthor thanksAmeyKolati, LennyNg, andBulentTosunforusefuldis-
cussionsande-mailexchangesduringthepreparationofthiswork. HealsothanksThomasVogel
foragreeingtoallowthe inclusion ofTheorem2.2thatheandthe authorworked outsome years
ago. The author also thanks Hansjo¨rg Geiges and Mirko Klukas for interesting discussions con-
cerning the classification of loose Legendrianknots. The author is also gratefulto the refereesof
thispaperwhomadevaluablecommentsthathelpedclarifymanypointsinthepaper. Thiswork
waspartiallysupportedbyNSFgrantDMS-0804820.
2. BACKGROUND CONCEPTS
WeassumethereaderisfamiliarwithconvexsurfacetheoryandLegendrianknots, seeforex-
ample [13, 16]. For convenience we recallsome of the keyfeaturesof Legendrianand transverse
knots used in this paper in Subsection 2.1. In the following subsection we recall Eliashberg and
Fraser’s classification of non-loose Legendrian unknots. We sketch a simple proof of this result
thatT.Vogelandtheauthorhadworkedout,andobservetheimmediatecorollaryconcerningnon-
loose transverse unknots. In Subsetction 2.3 we recall the definition of Giroux torsion and make
severalobservationsnecessaryfor the proofsof our main results. In the last subsection we recall
thenotionofquasi-convexityintroducedin[18].
2.1. NeighborhoodsofLegendrianandtransverseknots. RecallthataconvextorusT inacontact
manifold(M,ξ)willhaveanevennumberofdividingcurvesofsomeslope.Wecalltheslopeofthe
dividingcurvesonaconvextorusthedividingslopeofthetorus. Iftherearejusttwodividingcurves
then using the Legendrian realization principle of Giroux we can arrange that the characteristic
foliation has two lines of singularities parallel to the dividing curves, these are called Legendrian
divides, and the rest of the foliation is by curves of some slope not equal to the dividing slope.
These curves are called ruling curves and their slope is called the ruling slope. Any convex torus
withsuch acharacteristicfoliation willbesaid tobeinstandardform. Note thatgivenatorus in
ONKNOTSINOVERTWISTEDCONTACTSTRUCTURES 9
standardformwecanperturbthefoliationtohavetwoclosedleavesparalleltothedividingcurves
andtheotherleavesspiralingfromoneclosedleaftotheother.
Theregularneighborhood theoremforLegendriansubmanifoldssaysthatgivenaLegendrian
knotLinacontactmanifold(M,ξ)thereissomeneighborhoodN ofLthatiscontactomorphicto
aneighborhoodN oftheimageofthex-axisinR3/(x 7→ x+1) ∼= S1×R2 withcontactstructure
′
ξ = ker(dz −ydx). Byshrinking N andN ifnecessarywecanassume thatN isa diskinthe
std ′ ′
yz-planetimes the image of the x-axis. It is easyto see, using the model N , that ∂N is a convex
′
torus with two dividing curves of slope 1 where n = tb(L). Thus we can assume that ∂N is in
n
standardform. Moreover,noticethatL = {(x,±ǫ,0)}⊂ N isa(±)-transversecurve. Theimage
′
±
ofL inN iscalledthetransversepush-offofLandL iscalledthenegativetransversepush-off. One
+
−
mayeasilycheckthatL iswell-definedandcomputethat
±
sl(L )=tb(L)∓r(L).
±
WenowrecallhowtounderstandstabilizationsanddestabilizationsofaLegendrianknotK in
termsofthestandardneighborhood. Insidethestandardneighborhood N ofLwecanpositively
ornegativelystabilizeL.DenotetheresultS (L),respectivelyS (L). LetN beaneighborhoodof
+ s
−
thestabilizationofLinsideN. AsabovewecanassumethatN hasconvexboundaryinstandard
s
form. Itwillhavedividingslope 1 .ThustheregionN\N isdiffeomorphictoT2×[0,1]andthe
n 1 s
contactstructureonitiseasilyseen−tobea“basicslice”,see[24]. Thereareexactlytwobasicslices
withgivendividingcurvesontheirboundaryandastherearetwotypesofstabilizationofLwesee
thatthebasicsliceN\N isdeterminedbythetypeofstabilizationdone,andviceversa.Moreover
s
ifN isastandardneighborhoodofLthenLdestabilizesifthesolidtorusN canbethickenedtoa
solidtorusN withconvexboundaryinstandardformwithdividingslope 1 . Moreoverthesign
d n+1
ofthedestabilizationwillbedeterminedbythebasicsliceN \N.
d
DenotebyS thesolidtorus{(φ,(r,θ))|r ≤ a} ⊂ S1×R2,where(r,θ)arepolarcoordinateson
a
R2 andφistheangularcoordinateonS1,withthecontactstructureξ = ker(dφ+r2dθ). Given
cyl
atransverseknotK inacontactmanifold(M,ξ)onemayuseastandardMosertypeargumentto
showthatthereisaneighborhood N ofK inM andsome positivenumber asuchthat(N,ξ| )is
ξ
contactomorphictoS . Noticethatthetori∂S insideofS havelinearcharacteristicfoliationsof
a b a
slope −b2. Thusfor allintegerswith 1 < awe havetoriT = ∂S with linearcharacteristic
√n n 1/√n
foliationofslope−1.LetL bealeafofthecharacteristicfoliationofT . ClearlyL isaLegendrian
n n n n
curveinthesameknottypeasT andtb(L )= −n. AnyLegendrianLLegendrianisotopictoone
n
oftheL soconstructedwillbecalledaLegendrianapproximationofK.
n
Lemma2.1(Etnyre-Honda2001,[16]). IfL isaLegendrianapproximationofthetransverseknotKthen
n
(L ) istransverselyisotopictoK. Moreover,L isLegendrianisotopictothenegativestabilizationof
n + n+1
L . (cid:3)
n
2.2. Non-looseknotsinS3. Inthissectionwerecallthefollowingcoarseclassificationofunknots
in overtwisted contact structures on S3. To state the theorem we recall that the homotopy class
of a plane field ξ on M can be determined by computing two invariants [23]: the 2-dimensional
invariant,whichisdeterminedbythespinc structures associatedtotheplanefield(ifH2(M;Z)
ξ
hasno2-torsionthens isdeterminedbytheEulerclasse(ξ)ofξ)andthe3-dimensionalinvariant
ξ
d (ξ). Inparticular,onthe3–spherethe2-dimensionalinvariantofaplanefieldvanishesandsoit
3
isdeterminedbyits3-dimensionalinvariantd (ξ)∈Z.
3
Theorem2.2(Eliashberg-Fraser2009,[11]). ThereisauniqueovertwistedcontactstructureξonS3that
contains non-loose Legendrian unknots. This contact structure has d (ξ) = 1 and is shown in Figure 3.
3
EveryLegendrianunknotwithtightcomplementhastb > 0anduptocontactomorphismthereareexactly
twosuchknotswithtb=n>1,theyaredistinguishedbytheirrotationnumberswhichare±(n−1),and
auniquesuchknotwithtb=1.
WeincludeabriefproofofthisresultthattheauthorworkedoutwithThomasVogelbutnever
published. This is essentially the same as the proof in [11], though it is couched in somewhat
differentlanguageandweidentifyasurgerypicturefortheminimaltbexampleinFigure3.Before
giving the proof we notice two simple corollaries concerning transverse unknots and non-loose
unknotsinothercontactmanifolds.
10 JOHNB.ETNYRE
(+1)
(+1)
FIGURE 3. The two contact surgeries on the left give the contact structure ξ in
Theorem2.2. ThedottedLegendrianbecomesaLegendrianunknotLwithtb = 1
inthesurgeredmanifold. ThediskLboundsisindicatedontheright.
Corollary 2.3. Any transverse unknot in any overtwisted contact manifold is loose and hence coarsely
determinedbyitsself-linkingnumber.
Proof. Sinceanynon-looseLegendrianunknotcanbenegativelystabilizeduntilitisloose,Lemma2.1
impliesthatanytransverseunknotisthetransversepush-offofalooseLegendrian.Sincethetrans-
verse push-off can be done in any C -neighborhood of the Legendrian knot it is clear that the
∞
overtwisteddiskinthecomplementoftheLegendrianisalsointhecomplementofthetransverse
knot. Thus any transverse unknot in an overtwisted contact structure is loose. The rest follows
fromTheorem1.5 (cid:3)
Corollary2.4. Onafixed3-manifoldM withfixedspinc structuresthereisanovertwistedcontactstruc-
turehavingnon-looseunknotsandassociatedspincstructuresifandonlyifthereisatightcontactstructure
onM withassociatedspinc structures. Moreover,thenumberofsuchovertwistedcontactstructureassoci-
atedtosisequaltothenumberoftightcontactstructuresassociatedtos.
Proof. Let U be an unknot in M and U be an unknot in S3. The unique prime decomposition
′
of tight contact manifolds [2] implies that any tight contact structure on M \U comes fromtight
contactstructuresonM andS3\U . Theresultfollows. (cid:3)
′
Wenowturntotheclassificationofnon-looseunknotsinS3.
ProofofTheorem2.2. Letξ beanyovertwisted contactstructureonS3 andLa Legendrianunknot
in(S3,ξ). WecandecomposeS3as
S3 =V ∪ V
1 φ 2
whereV =S1×D2andφ:∂V →∂V isthemapgivenby
i 1 2
0 1
φ=
(cid:18)1 0(cid:19)
in meridian-longitude coordinates on each ∂V . Moreover we can assume that V is a standard
i 1
neighborhood of L with convex boundary having dividing slope 1/n where n = tb(L). If we
assumethecomplementofListightthenthecontactstructureonV istight(ofcourse,thecontact
2
structure on V is alwaystight). The boundaryof V is convex and has dividingslope n. Clearly
1 2
L,uptocontactomorphism, isdeterminedbythe contactstructureonV . In[21,24]tightcontact
2
structuresonsolidtoriwereclassifiedandthisleadstothefollowingpossibilitiesfortightcontact
structuresonV . Ifn<0thenthereareprecisely|n|distincttightcontactstructuresonV withthe
2 2
givendividingcurvesontheboundary.Allofthecontactstructuresarerealizedasthecomplement
of a tb = n, Legendrian unknot in the standard tight contact structure on S3. Thus they never
showupasthecomplementofanunknotinanovertwistedcontactstructure. Ifn =0thenV has
2
dividingslope0andhencethecontactstructuremustbeovertwisted.Thusn6=0ifthecomplement
of Lis tight. Finally, if n > 1 thentherearetwotight contactstructureson V andonly one tight
2
contact structure on V when n = 1. Thus we have shown there is a unique Legendrian knot in
2
anovertwistedcontactstructureonS3 thathastb = 1andthereareatmosttwowhentb > 1. We
arelefttoshowthatthesealloccurinthesamecontactstructure,thatcontactstructurehasd = 1
3