Selected Title s i n Thi s Subserie s

42 S . Tabachnikov , Editor , Differentia l an d Symplecti c Topolog y o f Knot s an d Curve s

(TRANS2/190)

41 V . Buslaev , M . Solomyak , an d D . Yafaev , Editors , Differentia l Operator s an d

Spectral Theor y (M . Sh . Birman' s 70t h anniversar y collection ) (TRANS2/189 )

40 M . V . Karasev , Editor , Coheren t Transform , Quantization , an d Poisso n Geometr y

(TRANS2/187)

39 A . Khovanskii , A . Varchenko , an d V . Vassiliev , Editors , Geometr y o f Differentia l

Equations (TRANS2/186 )

38 B . Feigi n an d V . Vassiliev , Editors , Topic s i n Quantu m Group s an d Finite-Typ e

Invariants (Mathematic s a t th e Independen t Universit y o f Moscow ) (TRANS2/185 )

37 Pete r Kuchmen t an d Vladimi r Lin , Editors , Voronez h Winte r Mathematica l School s

(Dedicated t o Seli m Krein ) (TRANS2/184 )

36 V . E . Zakharov , Editor , Nonlinea r Wave s an d Wea k Turbulenc e (TRANS2/182 )

35 G . I . Olshanski , Editor , Kirillov' s Semina r o n Representatio n Theor y (TRANS2/181 )

34 A . Khovanskii , A . Varchenko , an d V . Vassiliev , Editors , Topic s i n Singularit y

Theory (TRANS2/180 )

33 V . M . Buchstabe r an d S . P . Novikov , Editors , Solitons , Geometry , an d Topology : O n

the Crossroa d (TRANS2/179 )

32 R . L . Dobrushin , R . A . Minlos , M . A . Shubin , an d A . M . Vershik , Editors , Topics i n Statistica l an d Theoretica l Physic s (F . A . Berezi n Memoria l Volume )

(TRANS2/177)

31 R . L . Dobrushin , R . A . Minlos , M . A . Shubin , an d A . M . Vershik , Editors ,

Contemporary Mathematica l Physic s (F . A . Berezi n Memoria l Volume ) (TRANS2/175 )

30 A . A . Bolibruch , A . S . Merkur'ev , an d N . Yu . Ne t sve taev , Editors , Mathematic s

in St . Petersbur g (TRANS2/174 )

29 V . Kharlamov , A . Korchagin , G . Polotovskff , an d O . Viro , Editors , Topolog y o f

Real Algebrai c Varietie s an d Relate d Topic s (TRANS2/173 )

28 L . A . Bunimovich , B . M . Gurevich , an d Ya . B . Pesin , Editors , Sinai' s Mosco w

Seminar o n Dynamica l System s (TRANS2/171 )

27 S . P . Novikov , Editor , Topic s i n Topolog y an d Mathematica l Physic s (TRANS2/170 )

26 S . G . Gindiki n an d E . B . Vinberg , Editors , Li e Group s an d Li e Algebras : E . B .

Dynkin's Semina r (TRANS2/169 )

25 V . V . Kozlov , Editor , Dynamica l System s i n Classica l Mechanic s (TRANS2/168 )

24 V . V . Lychagin , Editor , Th e Interpla y betwee n Differentia l Geometr y an d Differentia l

Equations (TRANS2/167 )

23 Yu . Ilyashenk o an d S . Yakovenko , Editors , Concernin g th e Hilber t 16t h Proble m

(TRANS2/165)

22 N . N . Uraltseva , Editor , Nonlinea r Evolutio n Equation s (TRANS2/164 )

Published Earlie r a s Advance s i n Sovie t Mathematic s 21 V . I . Arnold , Editor , Singularitie s an d bifurcations , 199 4

20 R . L . Dobrushin , Editor , Probabilit y contribution s t o statistica l mechanics , 1994

19 V . A . Marchenko , Editor , Spectra l operato r theor y an d relate d topics , 199 4

18 Ole g Viro , Editor , Topolog y o f manifold s an d varieties , 199 4

17 D m i t r y Fuchs , Editor , Unconventiona l Li e algebras , 199 3

16 Serge i Gelfan d an d Simo n Gindikin , Editors , I . M . Gelfan d seminar , Part s 1 and 2 ,

1993

15 A . T . Fomenko , Editor , Minima l surfaces , 199 3

14 Yu . S . Il'yashenko , Editor , Nonlinea r Stoke s phenomena , 199 2

(See th e AM S catalo g fo r earlie r titles )

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Differential an d Symplecti c Topology o f Knot s and Curve s

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American Mathematica l Societ y

TRANSLATIONS Series 2 • Volum e 19 0

Advances in the Mathematical Sciences—42 (Formerly Advances in Soviet Mathematics)

Differential an d Symplecti c Topology o f Knot s and Curve s

S. Tabachniko v Editor

American Mathematical Societ y Providence, Rhode Island

° 3 5 D E D ^

http://dx.doi.org/10.1090/trans2/190

A D V A N C E S I N T H E M A T H E M A T I C A L S C I E N C E S

E D I T O R I A L C O M M I T T E E

V . I . A R N O L D

S. G . G I N D I K I N

V . P . M A S L O V

1991 Mathematics Subject Classification. P r i m a r y 57Gxx ; Secondar y 53Cxx .

ABSTRACT. Thi s collectio n o f paper s concern s th e topolog y an d geometr y o f knot s an d curves , especially i n th e presenc e o f contac t structures . Th e method s o f stud y var y fro m contemporar y knot theor y t o symplecti c an d contac t topology .

Library o f Congres s Car d Numbe r 91-64074 1 ISBN 0-8218-1354- 4

ISSN 0065-929 0

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Contents

Preface i x

Contact Topology , Tau t Immersions , an d Hilbert' s Fourt h Proble m JUAN CARLO S ALVARE Z PAIV A 1

On Legendr e Cobordism s EMMANUEL FERRAN D 2 3

Vassiliev Invariant s o f Knot s i n R 3 an d i n a Soli d Toru s VICTOR GORYUNO V 3 7

Finite Typ e Invariant s o f Generi c Immersion s o f M n int o R 2n ar e Trivia l TADEUSZ JANUSZKIEWIC Z AN D JACE K SWIATKOWSK I 6 1

On Enumeratio n o f Unicursa l Curve s SERGEI K . LAND O 7 7

Vassiliev Invariant s Classif y Fla t Braid s ALEXANDER B . MERKO V 8 3

New Whitney-type Formula s fo r Plan e Curve s MICHAEL POLYA K 10 3

Tree-Like Curve s an d Thei r Numbe r o f Inflectio n Point s BORIS SHAPIR O 11 3

Geometry o f Exac t Transvers e Lin e Field s an d Projectiv e Billiard s SERGE TABACHNIKO V 13 1

Shadows o f Wave Front s an d Arnold-Bennequi n Typ e Invariant s o f Front s on Surface s an d Orbifold s VLADIMIR TCHERNO V 15 3

A Unified Approac h t o th e Fou r Verte x Theorems . I MASAAKI UMEHAR A 18 5

A Unified Approac h t o th e Fou r Verte x Theorems . I I GUDLAUGUR THORBERGSSO N AN D MASAAK I UMEHAR A 22 9

Topology o f Two-Connecte d Graph s an d Homolog y o f Space s o f Knot s VICTOR A . VASSILIE V 25 3

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Preface

In recent years we have been witnessing a revival of interest in topology of curves and knots . Thi s interest was stimulated b y progress in singularity theory, the adven t of "quantu m topology " an d th e theor y o f finite-typ e kno t invariants , an d als o b y the rapi d growt h o f symplecti c topolog y i n whic h curve s an d knot s appea r a s th e simplest, one-dimensiona l Lagrangia n an d Legendria n manifolds . Th e thre e areas , global singularitie s o f wav e front s an d caustics , finite-type invariant s o f knot s an d knot-like object s an d contac t geometr y an d topolog y o f Legendrian knots , defin e a rather wid e spectru m o f problems concerne d i n thi s volume .

It woul d b e appropriat e t o commen t o n th e wor d "curves " i n th e titl e o f thi s collection. Immerse d plan e curve s ca n b e treate d a s knot-lik e objects : a generi c curve ha s neithe r tripl e point s no r self-tangencies , bu t bot h type s o f singularitie s generically occu r i n one-paramete r familie s o f curve s (compar e t o curve s i n th e three-dimensional space , wher e doubl e point s genericall y appea r i n on e paramete r families). Thu s th e topologica l theorie s o f tripl e poin t fre e curve s an d o f non-self -tangent curve s ar e analogou s t o kno t theory . Thi s poin t o f vie w wa s pu t forwar d in th e recen t wor k o f th e Mosco w Singularit y Schoo l represente d i n thi s collectio n by abou t hal f o f the participants .

A smoot h plan e curv e ca n b e regarde d a s a Legendria n curv e i n th e contac t manifold o f contac t element s o f th e plane , an d curve s withou t self-tangencie s ar e Legendrian knots . I n thi s wa y differentia l geometr y o f curve s i s "embedded " i n contact geometr y o f Legendria n curves . A n exampl e muc h studie d i n recen t year s is th e classica l 4-verte x theore m b y S . Muckhopadhyay a (publishe d i n 1909) : th e curvature o f a close d conve x plan e curv e ha s a t leas t 4 extrema; thi s resul t ca n b e interpreted an d generalize d a s a theorem o f symplectic topology . Anothe r exampl e is th e Mobiu s theore m tha t a noncontractibl e embedde d curv e i n th e projectiv e plane has a t leas t thre e distinc t inflectio n points . Thes e result s an d thei r numerou s generalizations ar e closel y relate d t o work s o f Sturm , Hurwitz , Kellog g an d other s on disconjugat e differentia l equations .

For th e convenienc e o f th e reader , w e give very brie f summarie s o f the papers . J. C . Alvare z extend s th e theor y o f taut immersion s fro m plan e curve s t o wav e

fronts i n a Desarguesia n surface , tha t is , th e plan e o r th e spher e wit h a Finsle r metric whos e geodesie s ar e straigh t line s o r grea t circles .

E. Ferrand' s pape r concern s severa l notion s o f Legendre cobordism ; on e o f th e applications t o Legendria n knot s i n th e standar d contac t 3-spac e i s a n inequalit y for a ne w contac t isotop y invariant , th e ribbo n Legendr e genus , simila r t o th e Bennequin inequalit y fo r th e self-linkin g o f a Legendria n knot .

ix

x PREFAC E

V. Goryuno v extend s th e constructio n o f the Kontsevic h integra l fro m 3-spac e to th e soli d toru s an d give s a descriptio n o f th e spac e o f finite-type invariant s o f knots an d frame d knot s therei n i n term s o f chord diagrams .

T. Januszkiewic z an d J . Swiatkowsk i defin e finite-type invariant s fo r immer -sions o f n-manifold s t o 2n-dimensiona l spac e an d prov e that , i n a certai n sense , these invariant s ar e trivial .

S. Land o give s a n uppe r boun d fo r th e numbe r o f isotopi c classe s o f one -component immerse d spherica l curves ; the techniqu e i s that o f matri x integration .

The mai n resul t o f A. Merkov' s pape r i s that finite-type invariant s classif y flat braids; th e latte r play s a role i n th e theor y o f triple poin t fre e plan e curve s simila r to tha t o f the usua l braid s i n kno t theory .

M. Polya k finds a famil y o f loca l formula s fo r th e windin g numbe r o f a n im -mersed plan e curv e generalizin g a classica l resul t b y H . Whitney ; simila r formula s are give n fo r th e "strangeness" , a first degre e invarian t o f tripl e poin t fre e curve s introduced b y Arnold .

B. Shapir o give s bounds fo r th e minima l numbe r o f inflection point s o f smoot h plane curve s withi n a difTeomorphis m class .

S. Tabachnikov studie s a class of transverse lin e fields along hypersurfaces gen -eralizing the field of normals in Euclidean space; the motivation for this study comes from th e theor y o f mathematica l billiards .

V. Tchernov' s pape r concern s kno t invariant s o f first degre e i n circl e fibrations and a n extensio n o f V. Turaev' s shado w technique s t o invariant s o f wave front s o n surfaces.

The tw o paper s b y M . Umehar a an d G . Thorbergsso n provid e a unifie d ap -proach t o many know n result s o f the 4-verte x theore m typ e (on e of the mos t recen t is the theore m b y E . Ghys : th e Schwarzia n derivativ e o f ever y projectiv e lin e dif -feomorphism ha s a t leas t 4 zeroes) .

V. Vassiliev' s pape r concern s a promisin g ne w approac h t o computin g homol -ogy o f th e spac e o f knot s (th e zero-dimensiona l homolog y bein g th e spac e o f kno t invariants).

In concludin g thi s brie f introduction , i t i s a pleasur e t o mentio n th e trul y international natur e o f thi s collection : th e countrie s o f affiliatio n o f th e author s include Belgium , France , Germany , Israel , Japan , Poland , Russia , Sweden , Unite d Kingdom, an d th e U.S . Al l paper s hav e bee n refereed , an d i t i s a pleasur e t o acknowledge th e valuabl e contribution s mad e b y th e anonymou s referee s i n th e preparation o f this volume .

V. Arnol d S. Tabachniko v