Quick viewing(Text Mode)

Problems in Low Dimensional Contact Topology

Problems in Low Dimensional Contact Topology

Aeia ahmtclS it n nentoa Press International and ciety So Mathematical American c

Nsp re yteAeia nttt fMteaisadNFGatDMS Grant NSF and of Institute American the by orted supp LN

Esp re npr yNFGatDMS Grant NSF by part in orted supp JE

otc tutrtgt eedinkocne convex Legendrian tight structure Contact phrases and words Key

rmr scnayD M D secondary R Primary Classication Subject Mathematics

oevr weakly a Moreover llable weakly implies llable strongly and structure contact

ealta ooopial lal otc tutr sas togyllable strongly a also is structure contact llable holomorphically a that Recall

ital overtwisted virtually

nvral ihor tight universally

ooopial lal o ooopial semillable holomorphically or llable holomorphically

togylbersrnl semillable strongly or llable strongly

ekylberwal semillable weakly or llable weakly

otc tutr a e b may structure contact

hsqeto rasdw nosvrlsbqetos u rtnt htatight a that note rst but questions sub several into down breaks question This

ftgtcnatsrcue ote admit they do structures contact tight of

hc aiod aetgtcnatsrcueadwa types what and structures contact tight have Which Question

xsec n y so otc structures contact of es typ and Existence

see ology top and see contact

ai oin in notions basic the with familiar e b to assumed is reader the this In

TreDmninlCnatGeometry Contact Dimensional Three I

icse here discussed

problems the of some of list a compiled who Sablo Josh and sessions problem

the during notes exceptional ok to who Kazez Will and Honda Ko thank to like

ewudalso would We list problem this write to us encouraging and sessions problem the

o running for Conference Georgia the of organizers the thank to like would We

hog da ihLgnra knots Legendrian with deal through

Sections and geometry contact dimensional three with deal through tions

Sec background some adds and sessions the from problems collects article This

a u yJh tyeadf ue nLgnra nt n otc contact and knots Legendrian on cused fo and Etnyre John by run was

ioxadf ue npolm ntredmninlcnattpooy h second the ology top contact dimensional three in problems on cused fo and Giroux

Emmanuel by run was rst The geometry contact concerning held were sions

uigte eri nentoa o lg ofrne w rbe ses problem two Conference ology Top International Georgia the During

onB tyeadLnadL Ng L Lenhard and Etnyre B John

o ology Top Contact Dimensional Low in Problems

oue Volume

MPSuisi dacdMathematics Advanced in Studies AMSIP

GogaItrainlTpooyConference ology Top International Georgia

emti o ology Top Geometric contact tight a into osed decomp e b can N M sum connected the on structure

contact tight any Moreover structure contact tight a admit not es do P

ercl httePicrehmlg peewt h osadr orientation nonstandard the with homology Poincare the that recall We

nwihteplldbc otc tutr s lable l is structure contact ledback pul the which in M of cover nite

steesome there is M on structure contact tight ly universal a is If Question

nese usinmgtbetefollowing the e b might question easier An true not is converse the that known is It

r nvra ytgtcnatsrcue llable l structures contact tight ly universal Are Question

esadben nvral ih rvrulyovertwisted virtually or tight universally eing b and sense

hr r okoncnetosbewe ih tutr iglbenany in llable eing b structure tight a etween b connections known no are There

udmna groups fundamental

fcusi scnetrdta l aiod aersdal nite residually have manifolds all that conjectured is it course Of G of subgroup

si h opeeto oeieidxnormal index nite some of complement the in is G of element nontrivial every if

srsdal nite residually is G group a that Recall overtwisted virtually or tight universally

either is structure contact tight any that prove to hard not and known is it

o n aiodwt eiulyiefnaetlgroup fundamental nite residually a with manifold any For sets these of one

utfl into fall must structures contact tight all if clear not is it but sets disjoint form

ti biu htuieslytgtadvrulyoetitdcnatstructures contact overtwisted virtually and tight universally that obvious is It

o a n eemn facnatsrcuei llable l is structure contact a if determine one can How Question

hwn htacnatsrcuei o llable not is structure contact a that showing

twudbeectn odohrmtod for ds metho other nd to exciting e b would It curvature ositive p with manifolds

to applies only and Lisca by initiated was technique This theory Witten

h antcnqeue oso otc tutr sntlbei eberg Seib is llable not is structure contact a show to used technique main The

srqie obesml connected simply e b to required is X when question this consider to

oi sms interesting most is it So vary to X of group fundamental the allow we if llings

aystrong many innitely always are there show to hard o to not is it Actually

pt lwpadblowdown and blowup to up M xed a of

X lings l weak or strong many nitely only exist there Do Question

nparticular In llings of

slbe ti neetn ocnie h ube n y e typ and er numb the consider to interesting is it llable is structure contact a If

ooopialyllable y Holomorphicall

togysmlcialyllable y symplecticall Strongly llable symplectically Weakly

togysmlcialysmlale semillabl y symplecticall Strongly semillable y symplecticall Weakly

j

Tight

see relations their and llability of notions the of discussion

o oethorough more a For llable strongly e b also must structures llable weakly

nwthat know we homology on However also see the on

lah erg Eliashb by found were examples such rst the llable strongly not are that

structures contact llable weakly and llable weakly not are that structures

ti nw htteeaetgtcontact tight are there that known is It tight is structure contact llable

ONB TYEADLNADL NG L LENHARD AND ETNYRE B JOHN oa ae aiod di ih otc structure contact tight a admit manifolds Haken l al Do Question

nte m ratqeto is question ortant imp Another

aiod di ih otc structure contact tight a admit manifolds

y r olic erb hyp all that however ossible p still is It foliations taut without

manifolds olic erb hyp are there that shown een b recently has it but structures

hmit ih contact tight into them erturb p could we true were this If foliations taut admit

manifolds olic erb hyp all that conjectured once was It structures olic erb hyp of

hr r okoncntutoso otc tutrsi terms in structures contact of constructions known no are There structures contact

ftight of existence the out ab known is little very manifolds olic erb hyp for As

n lal structures llable any

u o admitting not but structures contact tight admitting manifolds known rst the e b

hsrsl ol atclryitrsig cuete hs would these then ecause b interesting particularly e b would result This structures

contact tight admit do manifolds these that some by least at elieved b is It

di ih otc tutrsa ll al at structures contact tight admit

oteSietmnflswt invariants with manifolds Seifert the Do Question

ol eyitrsigt know to interesting very e b would

It structures contact llable weakly admit not do

Lsahssonta h efr aiod ihinvariants with manifolds Seifert the that shown has Lisca In structures

contact llable weakly admit not do that manifolds other several are There

invariants with S over manifold Seifert a is which P

is structure contact tight a admit to not known currently manifold only The

see e with spaces ered b

Seifert many on structures contact llable construct also can Gompf Moreover

i

i

x to equal or than less integer greatest the is c x b and c b e where

i

P

3 2 1 3

e if on structure contact llable strongly a is there then

3 2 1 3

2 1

invariants with ers b singular three with S over space ered b Seifert a e b

2 1

3 2 1

Let concern main our is here results ove ab the Given spaces ered b

3 2 1

a osrce togylbecnatsrcue nmn Seifert many on structures contact llable strongly constructed has Gompf

otc structures contact

nrtoa oooyshrs rcnie toglblt rvrulyovertwisted virtually or llability strong consider or spheres homology rational on

hs hncnieigtea v usinoecnf cus fo can one questions ove ab the considering when Thus b with manifold

oito nayirreducible any on foliation a such construct can Gabai Moreover structure

noatgtnfcwal eilbeaduieslytgt contact tight universally and semillable weakly fact in tight a into ed erturb p

e b may foliation Reebless a that shown have Thurston and erg Eliashb

hc r nvra ytgto ita yovertwisted ly virtual or tight ly universal are Which lable l are

Which structure contact tight admit spaces bered Seifert Which Question

http ftgtsrcue ote support they do structures tight of type What

structure contact tight a admit manifolds hyperbolic l al Do Question

emtiaincnetrte h anqeto rasit two into breaks question main the then Geometrization

foebeivstecnetrdpcueo aiod oigfo Thurstons from coming manifolds of picture conjectured the elieves b one If

xsec ftgtcnatsrcuew ilawy osdrirdcbemanifolds irreducible consider always will we structures contact tight of existence

oe o di n ih otc tutrs hndiscussing when so structure contact tight any admit not es do P M form the

n aiodof manifold any Thus N on structure contact tight a and M on structure

CONTACT DIMENSIONAL LOW IN PROBLEMS of results the generalize to ossible p e b then might it Problem answering

by facilitated greatly e b would This cut rst the understanding is cess pro this

in step hardest The b with manifolds on structures contact tight construct

eopoiin to ositions decomp these used have Matic and Kazez Honda them

uport supp that manifolds studying in useful very een b have ositions decomp Haken

fl ln nopesbesrae ni n rie tadson no fthreeballs of union disjoint a at arrives one until surfaces incompressible along ifold

famnfl sawyo ucsieyctigteman the cutting successively of way a is manifold a of decomposition Haken A

ogtalwrbonoecudueLgnra ugr n saetasto see transition state and surgery Legendrian use could one ound b lower a get To

ntesraebundle surface the on structures contact tight of er numb the on ound b er upp an

uvso ufc bete n ol pl h ouint rbe t obtain to Problem to solution the apply could one then er b surface a on curves

o foecudnraietedividing the normalize could one if Now bundle the of dromy mono the by related

are onents comp oundary b the on foliation characteristic or curves dividing the

and is result the er b convex a along it split bundle surface a on structure

ie ih contact tight a given obvious is bundles surface to Problem of application The

eea manifolds general

aiod ihHkndcm stos and ositions decomp Haken with manifolds

ufc ude vrcircles over bundles surface

hspolmhsrlvnet h lsicto ftgtcnatsrcue on structures contact tight of classication the to relevance has problem This

icl oudrtn h iuto hnteElrcasi o extremal not is class Euler the when situation the understand to dicult

more much seems It curves dividing the on hold conditions mild some and

g e extremal is structure contact the of class Euler relative the when

on structures contact tight classify Matic and Kazez Honda In

g genus of

sasurface a is where on structures contact tight Classify Problem

icto fcnatsrcue is structures contact of sication

peicpolmwoerslto ol aegetipotnefrteclas the for ortance imp great have would resolution whose problem ecic sp A

efr bee spaces ered b Seifert

on results further for See isotopy to up structures contact tight of set the

denotes M Tight where orientation on ending dep or Tight

example For spaces ered b Seifert some on known also is classication The

bundles

bundles Torus

spaces Lens

S S S

urnl otc tutrsaecasdo h olwn manifolds following the on classied are structures contact Currently

lsiytgtcnatsrcue na manifolds l al on structures contact tight Classify Problem

fcus h anpolmhr is here problem main the course Of

nqeesadclassication and Uniqueness

hsormi neethr gi srtoa oooyspheres homology rational is again here interest main our Thus theory

o r fta visfoliation avoids that of pro a for also see structures contact tight universally

i skonta hs aiod di semillable admit manifolds these that known is it Gabai and Thurston

skont aeadfo h oko lah erg Eliashb of work the from and Haken e b to known is b with manifold Any

ONB TYEADLNADL NG L LENHARD AND ETNYRE B JOHN n

T to isotopic is g f T that so M in edded emb contact e b can T

o which for n largest the is M manifold contact a in T torus a of torsion The

romdo ital vrwse otc structures contact overtwisted virtually on erformed p e b

nayeet ti o la htasmlrcntuto can construction similar a that clear not is it event any In T to transverse knots

sn h wsigo aiu Legendrian various of twisting the using in and elow b see T of torsion the

sn nivratcalled invariant an using in measured is twisting This nite is T along

tl aydrn ih otc tutrs n ed oko httetwisting the that know to needs one structures contact tight dierent many itely

r inn are there that conclude To structure contact tight ecial sp a with egin b we

opoetgtes ti m ratthat ortant imp is it tightness prove To dierent are they that and tight versally

hndigtioems rv htaltecnatsrcue bandaeuni are obtained structures contact the all that prove must one this doing When T

n

ws oenear more twist structure contact the making is one Intuitively T

with T replacing by structures contact tight many innitely tains

ob one Then on ordinate co the is z and T on ordinates co are y x

n

dy nz cos dx nz sin ker

where

T to contactomorphic d o orho neighb a has T torus incompressible

the which in structure contact tight universally ecial sp one constructing rst by

teiiyo nvral ih otc tutrswr constructed were structures contact tight universally of innity the In

e otc structures contact ted

oa aiod di nyieymn ita yovertwis ly virtual many nitely only admit manifolds l al Do Question

oeotaogte is them among Foremost question this of renements interesting many course of

hs euttea v usin ssaehsbenaseehwvr hr are there however answered een b has stated as question ove ab the results these

nlgtof light In er numb nite a is that mind in keep to ortant imp is it statement

this for structures contact tight many nitely only admits manifold atoroidal

oriented closed any that Honda and Giroux Colin by announced een b cently

re has it Moreover structures contact tight universally many innitely admits

i a e hw htaycoe retdirdcbetria manifold toroidal irreducible oriented closed any that shown een b has it In

tures

struc contact tight many nitely only admit manifolds Which Question

xmlw a tr ihtefloigmtvtoa question motivational following the with start can we example

For questions ecic sp more ask can we classication complete a of Short

mlmn hsmtod ouint rbe wudbehelpful e b would Problem to solution a d metho this implement

ntyn to trying In together structures contact tight gluing for transition state called

od a eeo damtod metho a ed develop has Honda dies o handleb the on structures tight together glue

ih tutrso nabtaymnfloewudhv oudrtn o to how understand to have would one manifold arbitrary an on structures tight

ounderstand to then problem this solve could one If dicult seems classication

actual an but oundary b its on curves dividing xed with dy o handleb a on tures

udo h ube ftgtcnatstruc contact tight of er numb the on ound b er upp an give to hard not is it

from techniques Using done een b has this genus of dies o handleb For

lsiytgtcnatsrcue nhn lebodies hand on structures contact tight Classify Problem

n ih rtwn oconsider to want rst might one

manifold a on structures contact tight understand to So invariants son

aebenvr sfli tdigmnfl o lg egbace oes Cas covers branched eg ology top manifold studying in useful very een b have

eopositions decomp These osition decomp Heegaard a called is manifold a of tation

aiodcnbeotie ygun w ade istgtesc represen a such together dies o handleb two gluing by obtained e b can manifold

Any decompositions Heegaard use can one manifold general a study To

TOPOLOGY CONTACT DIMENSIONAL LOW IN PROBLEMS hn bu h otc structure contact the about thing

ostednmc ftewo otc etrl a any say eld vector contact a of ow the of dynamics the Does Question

o example For

eetdi ovxvco ed soitdt ie otc structure contact given a to associated elds vector convex in reected

overtwistedness virtual and lability l like properties are How Question

saohrt lt td otc structures contact study to ol to another us

hsgives This convex is manifold closed a on structure contact any that from

ideas using proved has Giroux Recently M on eld vector contact like gradient

sa is there if convex called is structure contact The preserves v of ow the if

eld vector contact a called is M manifold contact a on v eld vector A

ovxcnatsrcue n pe kdcm ositions decomp ok o b en op and structures contact Convex

r hyncsaiycnatisotopic contact necessarily they Are maximal and topic

i

r mohyiso smoothly are M I T two Suppose Question

eae usinis question related A

i

T n M of pieces the on

fw aeaclassication a have we if M on structures contact classify we Can osition decomp

k

samnmlsse ftr ie yteJSJ the by given tori of system minimal a is T T that ose supp explicitly More

steeacnatJJdecomposition JSJ contact a there Is Question

M of osition decomp JSJ

hs oigv h JohannsonShalenJaco the give tori These erty prop this to ect resp with minimal

i

r tria rSiet rd oevr hscleto suiu fi is it if unique is collection this moreover ered b Seifert or atoroidal are T n M

k

uhta h opoet of onents comp the that such T T tori incompressible of collection a is there

M manifold orientable irreducible an given ecically Sp tori incompressible

et aiodi sal eopoe along osed decomp usually is manifold a Next see M Tight M Tight

M M Tight know we M M sum connected a for since structures contact

lyrsrcst reuil aiodOemyas oti hncnieigtight considering when this do also may One manifolds irreducible to restricts ally

issneayoinal aiodhsauiu rm eopoiin n usu one osition decomp prime unique a has manifold orientable any since First

ways various in them oses decomp usually one manifolds studying When

a ag oso oeie ml o ling l no imply sometimes torsion large Can Question

peicly eask we ecically Sp case the always this Is llable weakly only are rest the

ihmnmltrinaesrnl lalwhile llable strongly are torsion minimal with structures contact tight the only

w nwthat know we T on structures contact tight the eg examples ecic sp many In

o r oso n llblt related lability l and torsion are How Question

ecnas ask also can We ecial sp very are constructed structures contact the nite eing b

torsion on less or more rely which HondaKazezMatic and Colin of theorems

inniteness the for that Note torsion nite have tight arbitrary an es do

o xmli edo h y tei httetrsi omli oi theorem Colins in normal is torus the that othesis hyp the drop we if example For

strinawy nt natgtcnatstructure contact tight a in nite always torsion Is Question

torsion

minimal with structures contact many nitely only there Are Question

nwrtefloigquestions following the answer

n

ewudlk to like would We e b to dened is torsion the edded emb e b can such no if

ONB TYEADLNADL NG L LENHARD AND ETNYRE B JOHN hsmachinery this

aebenal oase oelnsadn usin bot rdlnsusing links ered b out ab questions longstanding some answer to able een b have

huhaseigti rbe ih emoel piitc ioxadG dman o Go and Giroux optimistic overly seem might problem this answering Though

ewe ovxcnatsrcue n pnbo decompositions book open and structures contact convex between

ioxscorrespondence Girouxs of applications topological purely Find Problem

ow n hsscinwt eyo neddproblem ended en op very a with section this end we So now time some quite for

pe kdcm stoshv e sdt td h o lg fmanifolds of ology top the study to used een b have ositions decomp ok o b en Op

pe kdcm sto n upt h otc homology contact the outputs and osition decomp ok o b en op

idno the of binding the out ab information ological top andor dromy mono the out ab

ewudlk osea cieagrtmta ae yaia information dynamical takes that algorithm eective an see to like would We data

ok o b en op the from computed e b can homology contact that clear is it osition p

ic otc tutr a eosrce rma dpe pe kdecom ok o b en op adapted an from reconstructed e b can structure contact a Since

otc tutr ecmue ntrso naatdoe okdecomposition book open adapted an of terms in computed be structure contact

a of theory eld symplectic or homology contact the Can Question

eas ask also We

M on etc structures tight ly universal l al

vra ih tutrsrover or structures tight l al over g of minimum the is What Question

of properties any

to related it Is structure contact the of invariant eective an it is so If tures

eeetvl optdfraycaso otc struc contact of class any for computed eectively be g Can Question

B in brations of F ers b all over taken

hr h iiu is minimum the where g F genus f min g Let adapted is which to ositions p

h e falo nboo decom ok o b en op all of set the e b B Let structure contact a and M Fix

ita yovertwisted ly Virtual tight ly Universal lable l ly symplectical strongly

or Weakly tight is structure contact adapted the that imply decomposition book

open an of map monodromy or link the on conditions What Question

ecnreQeto to Question rene can we

Thus twists Dehn ositive p of duct pro a is dromy mono whose osition decomp ok o b

hwta otc tutr sSenlbei n nyi ti dpe oa pen op an to adapted is it if only and if llable Stein is structure contact a that show

ioxcan Giroux also see Piergallini and Loi of work Generalizing

tutrsado nboo eopositions decomp ok o b en op and structures

contact convex etween b ondence corresp onetoone a is there Hence adapted is

ioxcncntuta pe kdcm sto owihtecnatstructure contact the which to osition decomp ok o b en op an construct can Giroux

oevr ie ovxcnatstructure contact convex a given Moreover structure contact convex adapted an

hsfo no nboo eopoiin n gets one osition decomp ok o b en op an from Thus ok o b en op the of ers b the to

hti transverse is that M on eld vector contact a construct to how shown has Giroux

oteo nboo eoposition decomp ok o b en op the to adapted e b to said is ker structure contact

The M over extend to hard not is it and L n M on form contact a to descends

that F on form contact a is t t t dt Then F on

savlm form volume a is d that such F on form a nd can one er Winkelnkemp

and Thurston Following monodromy the called F F dieomorphism a by

g f F to g f F gluing by F from obtained is L n M then L n M of

bration the of er b the is F If ok o b en op the of binding the called L link a of

sarto ftecomplement the of bration a is M manifold closed a of decomposition book open

a rvdda neetn praht usinRcl htan that Recall Question to approach interesting an provided has Giroux

TOPOLOGY CONTACT DIMENSIONAL LOW IN PROBLEMS mesdoetitdds htcnbeuwudi cover a in unwound e b can that disk overtwisted immersed

h esni htteei oflk nntbonigan ounding b unknot Hopflike a is there that is reason The overtwisted virtually

is structure contact the then cover nite a in disk edded emb an to lifts Figure

ugr naLgnra nta nFgr adteimre iko h etof left the on disk immersed the and Figure in as knot Legendrian a on surgery

a hw hti otc tutr sotie rmLegendrian from obtained is structure contact a if that shown has Gompf

Miscellaneous

level the at true

hsi only is this structures contact overtwisted with ondence corresp onetoone in are

oeta lah r hoe oe o tt hthmtp lse fpaeelds plane of classes that state not es do theorem ergs Eliashb that Note

wse ik hc r o otc isotopic contact not are which disks twisted

osteeeita vrwse otc tutr ihtoover two with structure contact overtwisted an exist there Does Question

oemgts att osdrtefollowing the consider to want rst might one

using Problem address to order In elements two least at contains structure

contact overtwisted an for that shown een b has it In directly problem

this address to dicult it make which in theorems the of most in otheses p

hy technical are there but here helpful very e b should in work ergs Eliashb

sovertwisted is when M Di Compute Problem

elements n least at contains on inclusion the of kernel the

n

T for that know also We structure contact a by out picked is twisting of

direction preferred a since T on structures contact tight for on surjective not

h nlso is inclusion The groups homotopy higher out ab say one can What level the

hsmpi nioopimon isomorphism an is map this structure contact tight standard the with S For

foinainpeevn dieomorphisms orientationpreserving of

h group the M Di to M Di from map inclusion the Study Problem

M Di Compute Problem

eea neetn upolm are subproblems interesting Several

of contactomorphisms of

M Di group the understand M on structure contact a Given Problem

ie otc structure contact given

a of contactomorphisms of group the concerning known little very is There

Contactomorphisms

hc t wsignm ri is er numb twisting its which

to ect resp with right the on shown as disk immersed an ounds b

curve darker The structure contact overtwisted virtually a yield

eedinsreyo h ihe uvso h etcan left the on curves lighter the on surgery Legendrian Figure

ONB TYEADLNADL NG L LENHARD AND ETNYRE B JOHN nadtofralof all for addition In es typ knot prime to attention restrict usually will we elow b

hs nordiscussions our in Thus Section see summands prime the for classication

the know we if e typ knot nonprime a in knots Legendrian classify also can We

std

S in knot eight gure the

structure contact tight any in knots torus

structure contact tight any in unknot the

simple

Legendrian are classify presently can we which es typ knot prime the of All

a achieved e b can r tb of values

eedinsml ntcasaini ocue ycluaigpeieywhich precisely calculating by concluded is classication knots simple Legendrian

For isotopic Legendrian e b must invariants classical equal with e typ this of knots

ftoLegendrian two if simple Legendrian called is e typ knot A r and tb knots drian

std

rcl htteeaetolsia nains fLegen of invariants classical two are there that recall S case the In

isotopy

pt Legendrian to up K L in knots Legendrian the classify K Given Problem

stefollowing the is

h anpolmi eedinko theory knot Legendrian in problem main The K e typ of knots Legendrian of set

eoethe denote K L let manifold contact any in e typ knot ological top a is K If

Examples

n adesapiain fteekost otc emtyadtpology top and geometry contact to knots these of applications address and

elwt h rbe fcasfigLgnra n rnvrekos etos Sections knots transverse and Legendrian classifying of problem the with deal

std

etosthrough Sections S e b to assumed is manifold contact ambient the stated

nesotherwise Unless knots transverse and Legendrian concerning denitions ductory

nti oue o intro for volume this in elsewhere Etnyre by article the to refer Please

eedinKnots Legendrian I I

uztitwihyed ih structure tight a yields which Lutztwist

un an there is tube overtwisted an has structure contact a If Question

vrwse iki culyebedded emb actually is disk overtwisted

oeepehp h uaieoetitdtbei nyimrewieeach while immersed only is e tub overtwisted putative the erhaps p However ture

struc contact tight a ort supp not es do P that fact the given unlikely is This

tube

a noetitdds lasb opee oa overtwisted an to completed be always disk overtwisted an Can Question

hsrie h aua question natural the raises This duced pro is disks

fovertwisted of e tub a structure contact a on twist Lutz a erform p we When

tutrsvasm ugry construction e surgerytyp some via structures

adaohrwy r l otc tutrsotie rmuieslytgtcontact tight universally from obtained structures contact all are way another Said

rcdr oda soitduiesll ih structure tight ly universal associated an nd to procedure

some there is structure contact overtwisted ly virtual a Given Question

n ol r oadestefloigquestion following the address to try could one

ftevrulyoetitdpeoeo ol ucetywl nesood o understo well suciently e b could phenomenon overtwisted virtually the If

wse otc structure contact twisted

a eawy n uha mesdds navrull over ly virtual a in disk immersed an such nd always we Can Question

TOPOLOGY CONTACT DIMENSIONAL LOW IN PROBLEMS where K L in knots Legendrian Classify Problem

oepriual m ratsubproblems ortant imp particularly some

mention to like would we problem Legendrian main the attack to trying In

tbeisotopy stable

stp seuvln ocasfigLgnra nt pto up knots Legendrian classifying to equivalent is isotopy transverse to up knots

tflosta lsiyn transverse classifying that follows It eg see n some for isotopic are

K S K S ie isotopic stably are K K knots Legendrian the if only and if

n n

isotopic transverse are K K knots transverse two then knots Legendrian

tblzto on stabilization negative and ositive p of erations op the S by denote we if Now

obtained thus knot a to isotopic transverse is knot transverse any Conversely

in eg used frequently also is choice other the but ectives ersp p many from

natural seems here given one The convention sign the to as literature the in

Nt hr ssm discrepancy some is there Note K on orientation the with agreesdisagrees knots

stransverse as K and K on induced orientation the that so eld contact the to

ypsigteko o tefi ieto tangent direction a in itself of o knot the pushing by K and K knots transverse

two obtain we K knot Legendrian oriented an Given knots Legendrian with

ute eut bottases nt emlkl oaiefo hi connection their from arise to likely seem knots transverse out ab results Further

ocntutfmle fkoswihaenttasesl simple transversely not are which knots of families construct to

aeaatddutrslsi ri theory braid in results dicult adapted have Menasco and Birman knots torus

nld l iterated all include these Menasco of work by simple transversely are knots

aepoe htalsaldecag reducible exchange socalled all that proved have Wrinkle and Birman foliations

braid using approach dierent completely a Through simple transversely are

aesonta ou nt n h gr ih knot eight gure the and knots torus that shown have EtnyreHonda and

iialEtnyre similarly simple transversely is unknot the that showed erg Eliashb

sn hrceitcflaintechniques foliation characteristic Using er numb selflinking their by determined

faltases nt nti nttpeaecompletely are e typ knot this in knots transverse all if simple transversely e typ knot

ala Call knots transverse of situation the discuss to moment a for digress We

eg from terminology knot usual the use we Here simple

Legendrian not is hence pages few a Figure in shown knot the instance

For Section in see will we as simple Legendrian are es typ knot all Not

article Etnyres see

lsictos eyo h wru hoyo ovxsrae eeo db Giroux by ed develop surfaces convex of theory owerful p the on rely classications

adpoal future probably and of results classication The isotopy Legendrian to

peain up erations op welldened are S stabilizations the that show to straightforward

is It Figure see knot the of jection pro z y front the to zigzags adds which

y a bandfo aiabko hog h peaino stabilization of eration op the through knot maximaltb a from obtained e b can e typ

h boeko y s n eedinko hc oe o aiiet nisknot its in tb maximize not es do which knot Legendrian any es typ knot ove ab the

by r change which S stabilizations dierent two are

There shown as zigzag a with jection pro z y front knots the of

tblzto faLgnra ntrpae segment a replaces knot Legendrian a of Stabilization Figure

− S

+ S

ONB TYEADLNADL NG L LENHARD AND ETNYRE B JOHN eunieult satmtclyl ose lo automatically is inequality nequin

Ben the violates that knot Legendrian any that note we remark nal a As

n not one

loose one r and tb with unknots Legendrian two exactly are there and

sloose is r or tb with in unknot Legendrian Any Conjecture

hna eedinukosaeloose are unknots Legendrian l al then

besides S on structure contact overtwisted any is If Conjecture

tutr yapyn n uztwist Lutz one applying by structure

bandfo h ih contact tight the from obtained S on structure contact the e b let example For

loose K L in knots

r llLegendrian l al are K types knot and manifolds contact what For Question

hsi ol neetn oknow to interesting e b would it Thus isotopic Legendrian

hnte are they then r and tb and e typ knot same the have knots Legendrian ose lo two

If knots Legendrian ose nonlo also are there but ose lo are structures contact

novertwisted in knots Legendrian most that seems It overtwisted is L of plement

com the on structure contact the if loose structure contact overtwisted an in K

alaLgnra knot Legendrian a Call structures contact overtwisted in knots Legendrian out ab

u itei known is little but structures contact tight in knots Legendrian classifying wards

a e aeto made een b has progress much that fact surprising a is It Problem

r and tb their by determined are and these of

r stabilizations are realizing knots Legendrian other l Al r and tb with type

knot the realizing knots Legendrian two exactly are There Conjecture

nttpe typ knot the is

h otovosoet consider to one obvious most The K L classify can we which for simple endrian

htaentLeg not are that K es typ knot prime no currently are There Problem

bee y r lcko hs oooysest interesting e b to seems holonomy whose knot olic erb hyp ered b

hsi a is this knot the at oking lo suggested has Kazez W ered b and olic erb hyp

oeei ih airt r nttpe htaeboth b are that es typ knot try to easier e b might it However e typ knot olic erb hyp

npriuai ol ra ohv eedincasaini obered nonb a in classication Legendrian a have to great e b would it particular In

y es typ knot olic erb hyp in knots Legendrian more some understanding from come

neato could interaction this to clues some and geometry contact and olic erb hyp tween

smnindi eto i ol eyhlflt nesaditrcin e b interactions understand to helpful very e b would it Section in mentioned As

ino y r lckos ofr eol nesadLgnra gr ih knots eight gure Legendrian understand only we far so knots olic erb hyp of tion

ecretyko eyltl botteLgnra classica Legendrian the out ab little very know currently We Problem

htsm e da ilbenee stegnso h ntgrows knot the of genus the as needed e b will ideas new some that

a sdi hsstaintlatsmtmsbti seems it but sometimes least at situation this in used e b can in techniques

ti rbbeta the that probable is It knots ered b genus at ok lo to is step next The classied

een b have es typ knot these of all in knots Legendrian and knots trefoil handed

h ny rdkoso eu r h gr ih ntadtergt n left and right the and knot eight gure the are genus of knots ered b only the

ent that note We elow b see knots nonprime for simple Legendrian always are knots

ti urnl sil tog o iey htpie ered b prime that likely not though ossible p currently is It Problem

eeaesm eak botteesubproblems these out ab remarks some are Here

sayko yei n vrwse otc structure contact overtwisted any in type knot any is K

sntLgnra simple Legendrian not is K

sahproi knot hyperbolic a is K

saee knot bered a is K

TOPOLOGY CONTACT DIMENSIONAL LOW IN PROBLEMS eeaesm te neetn tutrlquestions structural interesting other some are Here

r dierent but tb same the with unknots doubling by

xmlssmlrt h ostpckoso iuecnbeconstructed e b can Figure of knots nonisotopic the to similar Examples r and

aetb same the have doubles the then tb same the have knots original the if knots

edduln w ostpcLgnra nt ed onniooi Legendrian nonisotopic to leads knots Legendrian nonisotopic two doubling head

ti luil htWhite that plausible is It eg see satellites Legendrian on background For

orohrfvrt oooia bs fknots of abuse topological favorite other your

uauisums Murasugi

htha obigo oeohrstllt construction lite satel other some or doubling Whitehead

yeotie from obtained type

knot a in are that knots Legendrian about say one can What Question

a romo knots on erform p can

nadto ocnetdsmadcbig hr r ayohro rtosone erations op other many are there cabling and sum connected to addition In

pq

K L of terms in K L in knots Legendrian Classify Problem

hudalwoet drs h following the address to one allow should in techniques The curve q p a as

pq

K of representative a of d o orho neighb tubular a of oundary b the on sits K of

pq

htiarepresentative a is that K of cable q p the e b K let K e typ knot a Give

tb negative arbitrarily have but destabilize

steeako yewt eedinrpeettvsta onot do that representatives Legendrian with type knot a there Is Question

ceewudhv eldute ftease otefloigqeto eeyes were question following the to answer the if diculties real have would scheme

This these classify to then and knots tb maximal to destabilize knots Legendrian

st show to is results classication current all in used d metho the since ortant imp is

Question answered een b has question the which for es typ knot all for

yes is it but general in yes is question this to answer the that unlikely seems It

o hi nttype knot their for tb possible

oa eedinkosdsaiieutlte ec h maximal the reach they until destabilize knots Legendrian l al Do Question

tblztoti asst y by tb raises this stabilization

eoigthe removing then and knot Legendrian another of stabilization a ecome b to it

isotoping ie knot Legendrian a destabilizing is here notion ortant imp One K

o eea nttpes typ knot general for K L of structure the out ab known is nothing Virtually

K L classify completely can we which for K es typ knot ered b Legendriansimple

n h lsicto fLgnra eaietrskos n a aiydnon nd easily may one knots torus negative Legendrian of classication the and

result this Using K L and K L of terms in K K L classify can we Thus

K K if L L L L

L L S L S L

so w y es typ two of is relation equivalence the where

K K L

K L K L

once u fLgnra nt ie bijection a gives knots Legendrian of sum connected

ti hw that shown is it In eg see way standard a in L L sum connected

their form can we K L L and K L L knots Legendrian two Given

rttpclsrcuersl o eedinkoscnen once sums connected concerns knots Legendrian for result structure prototypical A

tutr Results Structure

ONB TYEADLNADL NG L LENHARD AND ETNYRE B JOHN in rened een b subsequently has DGA

the of formulation The isomorphism tame stable called equivalence algebraic

an under DGAs equivalent have they if only isotopic then are knots Legendrian

Two knot Legendrian a of diagram knot the given explicitly invariant DGA the

eea hoyo otc oooy hknvsdiinalw st rt down write to us allows denition Chekanovs homology contact of theory general

nadtoi a rae opeeycmiaoily ihu eeec othe to reference without combinatorially completely treated e b can it addition in

Chekanov to due formulation combinatorial simple a has homology contact

relative this general in compute to dicult is homology contact Although

eaiet h eedinko see knot Legendrian the to relative S of homology contact the precisely is

the of homology the dierential and grading a with algebra unital commutative

non free a DGA algebra graded dierential a of form the takes invariant This

ocnatmanifolds contact to techniques holomorphiccurve Gromovs applies which

nain fLgnra ntmtvtdb h eeomn fcnathomology contact of development the by motivated knots Legendrian of invariant

owerful p new a is knots two the etween b distinguish to used d metho The

lsia nainsaesoni iue Figure in shown are invariants classical

ihequal with knots Legendrian nonisotopic two the is e typ knot the efore b

mentioned As Hofer and erg Eliashb and Chekanov by endently indep

h rteapeo nttpewihi o eedinsml a r duced pro was simple Legendrian not is which e typ knot a of example rst The

eedinivrat n otc homology contact and invariants Legendrian

aetoLgnra nt stpc ntrso h oooyo h nttype knot the of topology the of terms in isotopic knots Legendrian two make

to necessary stabilizations of number the on bounds there Are Question

btoecnask can one but large arbitrarily e b can stabilizations necessary of er numb

the that sees one knots torus negative of sums connected ologically top are that

ycnieigLgnra knots Legendrian considering By stabilizations negative and ositive p oth b need

Nt htwe that Note n n some for isotopic Legendrian K S S and K S S

2 1 2 1

n n n n

ihietclcasclivratms have must invariants classical identical with K and K knots Legendrian topic

iso ologically top two that states Tabachnikov and Fuchs of result A

n a hwuigtecaatrsi ler seScin Section see algebra characteristic the using show can one

r xmlsi hc hsi o ucet as sucient not is this which in examples are there However examples many in

hsmv em osuce to seems move This Figure in given is Ng L by suggested move basic A

K

to K send l wil isotopy Legendrian with along which moves local of set a there

is tb same the with isotopic ly topological are K and K If Question

rsigchanges crossing

under changes tb of value maximal the how for formula a there is example For

hne fteko type knot the of changes

o r h eedinkosi nttp ce ycrossing by aected type knot a in knots Legendrian the are How Question

eesrl eediniooyclass isotopy Legendrian necessarily

ai oepeevn badtpooia lsbtnot but class ological top and tb preserving move Basic Figure

TOPOLOGY CONTACT DIMENSIONAL LOW IN PROBLEMS euti hsdirection this in result

o a for see ossible p seems following the fact In used e b already can technique

nain r urnl nwbeie xmlsi hc h ona olynomial Poincarep the which in examples esides b known currently are invariant

oapiain fthis of applications No fronts of ositions decomp admissible using knots Legendrian

aedsoee nte olsia nain of invariant nonclassical another discovered have Pushkar and Chekanov

isomorphic

tame stable they are homology same the have DGAs two If Question

ln hs iei ol atclrync oanswer to nice particularly e b would it lines these Along

rmteDGA the from

derived be can which invariants useful other there are particular In morphism

iso tame stable modulo DGAs of space the understand ly Ful Question

tb and K on only depends it precisely More invariant

topological a is algebra characteristic abelianized The Ng Conjecture

sil httecaatrsi ler sas ikdt o ology top to linked also is algebra characteristic the that ossible p

tseems It grading of sort any using without distinguished knots including in

ute plctoso h hrceitcagbaaegiven are algebra characteristic the of applications Further z y x z y x

saiybeLgnra stpct t irr t mg ne h contactomorphism the under image its mirror its to isotopic Legendrian e b essarily

utnec must r with knot Legendrian a whether asks which Tabachnikov and

irrqeto fFuchs of question mirror Legendrian the negative the in answer to used e b can

This algebra characteristic the is invariant DGAderived involved more A

eedinrepresentatives Legendrian

o hc hr r rirrl aynonisotopic many arbitrarily are there which for r and tb and es typ knot exist there

that shown e b can it lines same the Along isotopic Legendrian not are

fteDA hknvue hs lnmast hwta h nt nFgr Figure in knots the that show to olynomials p these used Chekanov DGA the of

rdddmnino linearizations of graded the calculate which olynomials p e Poincaretyp

are easiest The DGA the from derived invariants simpler used have plications

ap directly DGA the manipulating of diculty the of ecause b practice In

wihaentLgnra isotopic Legendrian not are which r and

tb with and e typ ological top of oth b knots Legendrian two

of ergHofer Eliashb and Chekanov to due Example Figure

ONB TYEADLNADL NG L LENHARD AND ETNYRE B JOHN ipe rtsew s h following the ask we step rst simpler

elsi otyt ov rbe s fr takn hspolm sapossibly p a As problem this attacking efore b rst Problem solve to try to realistic

more probably is it Thus K of theory eld symplectic the dene would which

rmteinformation the from M of homology contact the compute certainly almost can

oeeone however K of DGA the from merely M of homology contact the determine

can one that unlikely seems it great e b would relation a such nding Though

K to associated

rmteDAinvariant DGA the from M of homology contact the calculate to way a there Is

K on surgery Legendrian performing by obtained structure contact lable l Stein the

std

be M let and S inside knot Legendrian a be K Let Problem

o background for

see manifold contact another duces pro which surgery Legendrian called K on

surgery K tb natural a is there M in K knot Legendrian a Given

tepsa esnbedenition reasonable a at attempts

ee h leri omls a eitdseveral resisted has formalism algebraic the Here case Legendrian the for needed

btti sfrfo h aefrterltv ypetcl theory eld symplectic relative the for case the from far is this but d o understo

oehtwell somewhat is theory eld symplectic absolute ehind b structure algebraic The

eedinknots Legendrian

for theory eld symplectic l ful the understand ly Combinatorial Problem

wihgnrlzscnathmlgt eedinknots Legendrian to homology contact generalizes which GiventalHofer

nte sil prahi oapytesmlci edter fEisberg Eliashb of theory eld symplectic the apply to is approach ossible p Another

information

btmr opiae aeltsmyec euseful de enco may satellites complicated more but stabilizations for variants

in interesting no yields double Whitehead the satellite ossible p simplest The

aelt osrcinwihi aycsspodc nt ihnnrva DGAs nontrivial with knots duce pro cases many in which constructions satellite

n sil praht nesadn nainsfrsaiiain st use to is stabilizations for invariants understanding to approach ossible p One

e eto Section see

hspolmi fpriua m rac ntyn oudrtn rnvreknots transverse understand to trying in ortance imp particular of is problem This

tions

o stabiliza for invariants contacthomologytype useful there Are Problem

nain sta tvnse o tblzdLgnra knots Legendrian stabilized for vanishes it that is invariant

std

DGA the with diculty remaining jor ma One S to return now We

onie ihalnrddPicrepoyoildrvdfo h DGA the from derived olynomial p Poincare linkgraded a with coincides

invariant this that indicate calculations current all links these for mysteriously

o ology top algebraic and functions generating using links of class particular a for

a endaohrnncasclinvariant nonclassical another dened has Traynor R S for that note We

ers b the to transverse structure contact a with surfaces orientable closed

over bundles circle for diculty more with and Traynor and Ng by structure

contact tight standard the with R S torus solid the for done een b has This

eedinkosi te otc manifolds contact other in knots Legendrian

for theories homology contact relative combinatorial Construct Problem

std

S

nlge fteDAter a eie o otc aiod te than other manifolds contact for derived e b can theory DGA the of Analogues

ona oyoil o h orsodn DGA corresponding the for polynomials Poincare

from deduced be can invariant ChekanovPushkar The Conjecture

TOPOLOGY CONTACT DIMENSIONAL LOW IN PROBLEMS K n S std

nvra ytight ly universal j K n S is tion 3

cQues cf stabilization a not is but tb nonmaximal has K If Question

K n S std

nvra ytight ly universal j K 3

std

n S Is S in knot Legendrian tb maximal a be K Let Question

eeaesm nlquestions nal some are Here

knot the of examples

aia tb maximal the considering by problem this on working egin b to like might One

eesrl isotopic necessarily

and Are respectively K and K on surgery Legendrian performing by tained

ob manifolds contact the be M and M Let isotopic not are which r

and tb same the with K K knots Legendrian two Consider Question

skonfrtecoe case closed the for known is

Nothing also see no is answer the oundary b with manifolds allow we If

osLgnra ugr rsretightness preserve surgery Legendrian Does Question

r ue nte lal structure llable another duces pro

structure contact llable a in knot a on surgery Legendrian that known is

It structures contact on surgery Legendrian of eect the address next We

ntter nta fSi ritnter odsigihteeexamples these distinguish to theory ergWitten Seib of instead theory knot

ol n s Legendrian use one Could n pq q p ound b which manifolds Stein of classes

r itnuse yteChern the by distinguished are structures contact the theory ergWitten Seib Via

bandb ugr ndsic eedinknots Legendrian distinct on surgery by obtained n pq q p sphere Brieskorn

the on structures tight are These way this in distinguished e b also can tures

fcnatstruc contact of examples LiscaMatic the if see to interesting e b would It

knots Legendrian out ab information by and eld

r eemndb hi oooycaso plane of class homotopy their by determined are S over bundles T on structures

oegnrly ih contact tight generally More const y x to isotopic curves among er numb

n

r itnuse ytemxmltwisting maximal the by distinguished are dy nz cos dx nz sin forms

n

ie yte the by given Z n Z R T manifolds contact tight the example For

hc xssi nbtnti h other the in not but one in exists which r and number

twisting or tb certain a of knot Legendrian a nding by distinguished be always

M manifold same the on and structures contact tight Can Question

oepeievrino hspolmmgtbegvnb h olwn question following the by given e b might problem this of version precise more A

itnus te otc tutrsuigLgnra knots Legendrian using structures contact other Distinguish Problem

eedinrtaseskoscnbecridb h otc structure contact the by carried e b can knots transverse or Legendrian

nec ae n netgtswa ot of sorts what investigates one case each In spheres homology and torus

nt aepae iia oei itnusigbewe otc tutrso the on structures contact etween b distinguishing in role similar a played have knots

ysuyn rnvreukos n Legendrian and unknots transverse studying by R contact exotic an of existence

the established famously Bennequin geometry contact to knots Legendrian

oapiain of applications to knots Legendrian of erties prop intrinsic from turn now We

eaint otc geometry contact to Relation

osi qa h aeeld base the equal it Does

I sams trivial almost is Is form simple a have M of homology contact the does knot

nPolm sotie rmasaiie Legendrian stabilized a from obtained is Problem in M If Question

ONB TYEADLNADL NG L LENHARD AND ETNYRE B JOHN lar

nparticu In K L at oking lo by invariants knot ological top e nonnitetyp extract

a one Can invariants e nitetyp ological top standard and r tb esides b knots drian

codn oFcsTbcnkv hr r oietpeivrat fLegen of invariants e nitetyp no are there FuchsTabachnikov to According

nteoehnadteHML n aa oyoil nteother the on polynomials Kauman and HOMFLY the and hand one the on

K tb and K L between relation the deeply more Understand Problem

twudas luiaigt consider to illuminating e b also would It

lxne omo hrtnnorm Thurston or norm Alexander

the and invariants knot Legendrian between relationships Find Problem

o start a for see problem related

eas aetefollowing the have also We problem this to relevant are and Questions

eaeLgnra ntivrat oteAeadrmodule Alexander the to invariants knot Legendrian Relate Problem

ecncnie o lgclpo risbeie lcnsa well as sliceness esides b erties prop ological top consider can We

tb has mirror its or it either which for

nti ihro re o ocratt knot a to concordant or order of either is knot A Etnyre Conjecture

n ih bet rv eut uha h following the as such results prove to able e b might One eg See

ocrigsieknots slice concerning

a eueLgnra nt oipoeo erv te results other reprove or improve to knots Legendrian use we Can Question

C in order innite of is K then K tb if theory knot

atr swltruhLegendrian through well as factors Z the of some see to us allows Rudolph of result

The order have knots amphichiral that fact the from factors Z Z the of some

n a e a least at see can One factors Z Z many countably even or any are there whether

unknown currently is It factors Z Z ossibly p and Z Z Z many countably of

hsi ietsum direct a is this C group concordance knot oth smo the consider vein this In

sntslice not is K then

K tb if that shown has qv Rudolph particular In results of

o survey a for see olynomial p Kauman and olynomial p HOMFLY genus slice

bandi em fthe of terms in obtained een b have tb for ounds b er upp then Since K of genus

the of terms in K all for K tb on ound b er upp an to extends d metho His

std

b tb satises S in unknot the that establishing to terminology dern mo in

reduces S on structure contact exotic an of existence the of of pro Bennequins

ntsetenx paragraph next the see nite

hsnm ris er numb This K e typ of knots Legendrian all over tb maximal the denote K tb

std

let K e typ knot a For S in knot Legendrian a of invariant Bennequin

Thurston the in ded enco information the through principally geometry contact

with as well as ology top with connections deep has theory knot Legendrian

eaint o ology top to Relation

spoiigacneto otetpooyo h knot the of ology top the to connection a providing as

nesadn hs usin ol epu nadesn usina well as Question addressing in helpful e b could questions these Understanding

tight K n S of cover

std

Z the is S in knot Legendrian tb maximal a is K If Question

ut mznMr elsialw a ask can we realistically More amazing quite

htwudbe b would that yes e b to questions these of answers the for ossible p is it Though

TOPOLOGY CONTACT DIMENSIONAL LOW IN PROBLEMS eetoi electronic

oMtJ M xr oueIM ICM Volume Extra DMV J Math Doc ology top contact in Invariants erg Eliashb Y

1996 Notices Res Math

Internat torus the on structure contact llable holomorphically Unique erg Eliashb Y

o ton

Pbiho eih Hous Perish or Publish NY Brook Stony mathematics modern in methods

Topological manifolds contact tight in knots transversal and Legendrian erg Eliashb Y

42 Grenoble

nIsFourier Inst Ann work Martinets J since years twenty manifolds Contact erg Eliashb Y

abig Cambridge

oII oado n hms dabig nvriyPress University Cambridge eds Thomas and Donaldson I I Vol Manifolds Dimensional

Low of Geometry applications its and discs holomorphic by Filling erg Eliashb Y

1 Math

nent J Internat dimension of manifolds Stein of characterization ological Top erg Eliashb Y

98

net Math Invent manifolds on structures contact overtwisted of Classication erg Eliashb Y

19 Geom

nGoa Anal Global Ann S on structures contact overtwisted in knots Legendrian Dymara K

1 Topol Geom Algebr

F igadH egs ypetclblt ftgtcnatsrcue ntrsbundles torus on structures contact tight of llability Symplectic Geiges H and Ding F

1 Topol Geom Algebr complements link of

O abc n MnuO culnsadohrieulte o h hrtnnorm Thurston the for inequalities other and McMullens On Mangum B and Dasbach O

nti preparation in knots

Y hknvadP uhaAnl orcs ojcueadivrat fLegendrian of invariants and conjecture cusp four Arnolds Pushkar P and Chekanov Y

oapear app to Math Invent links Legendrian of algebra Dierential Chekanov Y

34

cl om u Sup Norm Ecole Sci Ann tendues contact de structures des torsion la Sur Colin V

51 Grenoble Fourier

nInst Ann tendues contact de varietes les dans admissibles Dehn de Chirurgies Colin V

76 Helv Math

Comment torodales varietes les sur tendues contact de structures de innite Une Colin V

324 Math I Ser Paris Sci Acad R C

Cln hrrisdidc ne stpe eshee aslsvrte ecnattendues contact de varietes les dans spheres de isotopies et un dindice Chirurgies Colin V

138

no ah Math of Ann links oundary b to concordant are links all Not Orr K and chran Co T

Clgr n Dned rtrafrnneitneo atflaini preparation in foliations taut of existence non for Criteria Duneld N and Calegari D

55 Geom Di J knots simple transversally On Wrinkle N and Birman J

Bra n Mnso tblzto ntebadgoppernmathGT preprint groups braid the in Stabilization Menasco W and Birman J

107{108 Asterisque Pfa de equations et Entrelacements Bennequin D

5 Topol Geom surfaces Stein compact on brations Lefschetz Ozbagci B and Akbulut S

182 Math J Pacic structures contact on note A Matveev R and Akbulut S

References

lmn result plement

s eedinko hoyt erv h oooia ntcom knot topological the reprove to theory knot Legendrian Use Problem

Luecke and Gordon of ity

std

oeetetpooia ntcmlmn rbe eursteingenu the requires problem complement knot ological top the However S

on structures contact tight of classication ergs Eliashb using easy somewhat is

complement its by determined is knot Legendrian a that showing Finally

tb dierent vastly have can mirror

its and K eg mirroring to sensitive very are knots Legendrian that is theory

otpooia knot ological top to theory knot Legendrian applying for motivation ossible p One

ontyeinvariant nonnitetype a tb Is Question

ONB TYEADLNADL NG L LENHARD AND ETNYRE B JOHN aiodpernmathGT preprint manifolds

bee y r lc olic erb hyp ered b on structures contact Tight Matic G and Kazez W Honda K

mathGT

K odW aeadG aic nteGbiEisbrhrtnterm preprint theorem GabaiEliashbergThurston the On Matic G and Kazez W Honda K

2002

Not Res Math Int theory osition decomp Convex Matic G and Kazez W Honda K

mathGT

loaalbeas available also ear app to J Math Duke structures contact tight Gluing Honda K

55 Geom Di J I I structures contact tight of classication the On Honda K

4 Topol Geom I structures contact tight of classication the On Honda K

Princeton Press University Princeton Stud Math of Ann Manifolds el Hemp J

82 Math Invent manifolds symplectic in curves Pseudoholomorphic Gromov M

2 Soc

AeMath Amer J complements their by determined are Knots Luecke J and Gordon McA C

148 Math of Ann surfaces Stein of construction dy o Handleb Gompf R

E iox ak tSafrDcm rada h nvriyo eriMy May Georgia of University the at and er Decemb Stanford at Talks Giroux E

76 Helv Math Comment

E iox tutrsd otc u e aiesrese ece udsu n surface dune audessus cercles en brees varietes les sur contact de Structures Giroux E

141 Math Invent

E iox tutrsd otc ndmnintose iuctosdsfuleae esurfaces de feuilletages des bifurcations et trois dimension en contact de Structures Giroux E

66 Helv Math Comment contact de ologie top en Convexite Giroux E

hs r ceedings pro these

P hgii ShoebreO h lsicto ftgtcnatsrcueesweein elsewhere structures contact tight of classication the On Schonenburger S Ghiggini P

18 Geom Di J manifolds of ology top the and Foliations Gabai D

36 Topology space contact

D uh n Tbcnkv nainso eedinadtases nt ntestandard the in knots transverse and Legendrian of Invariants Tabachnikov S and Fuchs D

preprint

D uhCeaolah r nainso eedinkos xsec faugmentations of existence knots Legendrian of invariants erg ChekanovEliashb Fuchs D

130 Soc Math Amer Proc invariants olynomial p and knots Legendrian On Ferrand E

oaper loaalbea mathSG at available also ear app to Geom Sympl J

J tye NadJ al oeetoinain n nainso eedinknots Legendrian of invariants and orientations Coherent Sablo J and Ng L Etnyre J

mathSG

preprint sums connected I I geometry contact and Konts Honda K and Etnyre J

148

net Math Invent llings symplectic no with structures contact Tight Honda K and Etnyre J

1 Geom Sympl

J eight gure and knots torus I geometry contact and Knots Honda K and Etnyre J

153

no ah Math of Ann structures contact tight of nonexistence the On Honda K and Etnyre J

J tye nr utr etrso otc emty lehr nteepoceedings pro these in elsewhere geometry contact on lectures ductory Intro Etnyre J

2 Math Contemp Commun spaces lens on structures contact Tight Etnyre J

3 Topol Geom knots torus Transversal Etnyre J

88 Appl Top ology top lowdimensional in convexity Symplectic Etnyre J

201 Math J Pacic knots Legendrian of imations

J ptiD uhadM eeCeaolahegivrat n rnvreapprox transverse and invariants ChekanovEliashberg Meyer M and Fuchs D Epstein J

AeMtS rvdne Providence c So Math Amer Confoliations Thurston W and erg Eliashb Y

peilVlmPr I I Part Volume ecial Sp 2000 Anal Funct Geom Aviv Tel

GAFA theory eld symplectic to duction Intro Hofer H and Givental A erg Eliashb Y

ah ocPoiec Providence c So Math

R r LcueNts Amer Notes Lecture c Pro CRM PQ Montreal dynamics and topology ometry

Ge in knots Legendrian trivial ologically top of Classication Fraser M and erg Eliashb Y

TOPOLOGY CONTACT DIMENSIONAL LOW IN PROBLEMS httpalummiteduwwwng URL

ngalummitedu address Email

mrcnIsiueo ahmtcPl lo A CA Alto Palo Mathematics of Institute American

httpwwwmathupenneduetnyre URL

etnyremathupennedu address Email

nvriyo enyvna hldlha A PA Philadelphia Pennsylvania of University

20 J Math Hokkaido dies o handleb symplectic and surgery Contact Weinstein A

5 Topol Geom links legendrian for olynomials p function Generating Traynor L

52 Soc

rc mr Math Amer Proc forms contact of existence the On er Winkelnkemp H and Thurston W

oaper loaalbea mathGT as available also ear app to Math of Ann oundary b incompressible

W hrtn y r lcsrcue nmnfls Dfrain fmnflswith manifolds of Deformations I I I manifolds on structures olic erb Hyp Thurston W

25 Math J Turkish fourmanifolds symplectic on brations Torus Smith I

J al nainso eedinkosi icebnls npreparation in bundles circle in knots Legendrian of Invariants Sablo J

119 Math Invent

L uopA btuto osieesvacnatgoer n cascl ag theory gauge classical and geometry contact via sliceness to obstruction An Rudolph L

Pbiho eihPes ekly Berkeley Press Perish or Publish links and Knots Rolfsen D

R o rs SaehaadM tirsac announcement research Stein M and Shareshian J erts Rob R

74 Helv Math Comment

H haadK nSml iglrte n o lg fsmlcial ligmanifold lling symplectically of ology top and singularities Simple Ono K and Ohta H

L g h eedinstliecntutopernmathGT preprint construction satellite Legendrian The Ng L

mathGT

as available also ear app to Topology invariants Legendrian Computable Ng L

39 Math J Osaka

A oi oeo hrtnWnenepe osrcino otc om nmanifolds on forms contact of construction ers ThurstonWinkelnkemp on note A Mori A

K ihceRltv oooysltigo ieeta ler fLgnra ik preprint link Legendrian of algebra dierential of splitting homotopy Relative Mishachev K

5 Topol Geom knots transversal and knots torus iterated On Menasco W

R avyv aka tnod eebe er Decemb Stanford at talk Matveyev R

143 Math Invent

B of covers branched as oundary b with surfaces Stein Compact Piergallini R and Loi A

51 Geom Di J

C iigtnadS ak btutn ortrini h lsia ntcnodnegroup concordance knot classical the in fourtorsion Obstructing Naik S and Livingston C

129

net Math Invent invariants ergWitten Seib and structures contact Tight Matic G and Lisca P

23 Math J Turkish Conference

P icO ypetclnso aiodPoceig fhGooaGoero ology GeometryTop Gokova th of ceedings Pro manifolds of llings symplectic On Lisca P

2 Topol Geom curvature scalar ositive p and llings Symplectic Lisca P

129 Math Invent structures contact and oles Monop Mrowka T and Kronheimer P

5

om nl Geom Anal Comm torus the on structures contact tight of classication The Kanda Y

ONB TYEADLNADL NG L LENHARD AND ETNYRE B JOHN