UPPSALA DISSERTATIONS IN MATHEMATICS

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Knots and Surfaces in RealAlgebraic and Contact Geometry

Johan Björklund

Department of MathematicsUppsala University

UPPSALA 2011

Department of MathematicsUppsala University

UPPSALA 2011

List of papers

This thesis is based on the following papers, which are referred to in the textby their Roman numerals.

I Björklund, J. Real Algebraic Knots of Low DegreeJournal of Knot Theory and its Ramifications, in pressPreprint available at arxiv.org/abs/0905.4186v1

II Björklund, J. Encomplexed Brown Invariant of Real Algebraic Sur-faces in RP3

SubmittedPreprint available at arxiv.org/abs/1108.1566

III Björklund, J. Legendrian Contact Homology in the Product of aPunctured Riemann Surface and the Real LineSubmittedPreprint available at arxiv.org/abs/1108.1568

Reprints were made with permission from the publishers.

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.1 Real Algebraic Knot Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.2 Invariants in Real Algebraic Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3 Contact Geometry and Legendrian Contact Homology . . . . . . . . . . . . . . 10

2 Summary in Swedish (sammanfattning på svenska) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.1 Reellalgebraisk Knutteori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 Invarianter för Reellalgebraiska Ytor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3 Kontaktgeometri och Kontakthomologi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1. Introduction

This thesis is devoted to the study of knots and surfaces with additional geo-metric structure. The thesis consists of three papers. In Paper I, real algebraicrational knots in RP3 are studied up to rigid isotopy, and a complete rigid iso-topy classification is obtained for such knots of degree at most 5. In Paper II aninvariant for smooth topology, the so-called Brown invariant, is generalized togeneric parametrized real algebraic surfaces in RP3. Paper III concerns Leg-endrian knots in the contact manifold P×R, where P is a punctured Riemannsurface. To distinguish Legendrian submanifolds of contact manifolds thereexists an invariant called contact homology. This invariant is defined using ageometric description, which comes from symplectic field theory. In Paper IIIwe describe how to calculate this invariant in a combinatorial manner in thislow dimensional setting.

1.1 Real Algebraic Knot TheoryThe real algebraic counterpart to 1-manifolds are real algebraic curves, i.e., al-gebraic curves defined by polynomials with real coefficients. Just as a smoothknot in some space X is an embedding of S1 into X , a real algebraic knot isan injective real algebraic map of some real algebraic curve C into some realalgebraic space X . We recall that two smooth knots are considered to be iso-topic if there is a path in the space of smooth knots connecting them. Tworeal algebraic knots are then said to be rigidly isotopic if there exists a path inthe space of real algebraic knots connecting them. In Paper I we consider theproblem of classifying real algebraic knots in projective space up to rigid iso-topy. There are several kinds of algebraic curves, classified by genus. Therehas been work done by Mikhalkin and Orevkov [5] classifying real algebraicknots of degree at most 6 up to smooth isotopy for arbitrary genus. In Paper I,we restrict ourselves to rational curves γ in RP3, i.e., γ : RP1→ RP3. Sincethe maps from RP1 to RP3 are defined by polynomials, they have some in-nate degree. Two real algebraic knots of different degrees can never be rigidlyisotopic. Thus, we have a rigid isotopy classification problem in each degree.To show that two real algebraic knots of the same degree are not rigidly iso-topic, we need rigid isotopy invariants. All isotopy invariants from smoothknot theory are rigid isotopy invariants since any rigid isotopy is in particularan isotopy. Finding invariants to distinguish knots up to smooth isotopy hasbeen one of the central questions in knot theory with new research coming

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Figure 1.1. A transition, modeled by y2 = x2(x−α), from a non solitary double pointto a real double point. The constant α varies from −1 to 1.

daily. As an example, one of the hot areas in knot theory today is Khovanovhomology, a categorification of the Jones polynomial. However, there existreal algebraic knots of the same degree which are smoothly isotopic but notrigidly isotopic, thus necessitating additional invariants beyond those used insmooth knot theory. The only known invariant (which does not come fromsmooth isotopy or the degree) is the encomplexed writhe discovered by Viro[8]. This invariant uses the natural complexification CK of a real algebraicknot K ⊂ RP3, by simply extending the appropriate map f : RP1 → RP3 toa map C f : CP1 → CP3. Given a generic projection to a plane in RP3 weget some finite number of double points. In contrast to smooth knot theory,where each such double point arises from two transversal pieces of the knotintersecting, we also get solitary double points arising from two complex con-jugate sheets of the knot intersecting.

See Figure 1.1 for an illustration of a real double points becoming a solitarydouble point. By assigning appropriate values ±1 to these double points andthen summing them, an invariant is obtained. In Paper I it was proved that tworeal algebraic rational knots of degree d ≤ 5 are rigidly isotopic if, and onlyif, the two knots have coinciding values of their encomplexed writhe. It wasalso shown that this does not hold true for higher degrees by the constructionof a counterexample in degree d = 6. Modeling RP3 as a ball with antipodalpoints identified, we illustrate all real algebraic knots of degree d ≤ 5 up tosmooth isotopy and mirror image in Figure 1.2.

1.2 Invariants in Real Algebraic GeometryFollowing the philosophy of Viro in [8] we consider an invariant in the realalgebraic world to be encomplexed if it is a natural extension of an invariantfrom the world of smooth topology. The encomplexed writhe was one exam-

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Figure 1.2. All real algebraic rational knots of degree d ≤ 5, up to smooth isotopyand mirror image. Starting from the top left we have:a straight line (d = 1), a circle(d = 1), the twocrossing knot (d = 4), a long trefoil (d = 5) and the 53-knot (d = 5).We let d = k denote that the knot first appears in degree k.

ple of an invariant inspired from topology in the real algebraic world. Anotherinvariant that survives from the smooth topological world is the Whitney indexfor curves in the plane. We recall that an immersion is a map with injectivedifferential. Two immersions of curves in the plane lie in the same compo-nent of the space of immersions if, and only if, they have the same Whitneyindex. The Whitney index can be calculated from the self intersections in thecase of a generic immersed curve. It turns out that this notion survives to(parametrized) real algebraic curves of Type I, where a corresponding real al-gebraic/encomplexed Whitney index can be calculated from self intersections(both solitary and non-solitary), as proved by Viro [9]. In Paper II we studya similar situation concerning the space of generically immersed oriented sur-faces in R3. The Brown invariant is an invariant up to regular homotopy ofimmersed surfaces. The Brown invariant of an immersed surface can be de-fined using the self intersection of the surface as has been shown by Kirby andMelvin [6]. In their article they express the Brown invariant by constructingan auxiliary curve called the “pushoff“ on the boundary of a tubular neigh-bourhood of the self intersection together with a natural projection to the selfintersection with 4 points in the preimage of each point in the self intersection.The linking number between this pushoff and the self intersection is shown togive the Brown invariant.

In Paper II we gave a definition of an encomplexed Brown invariant, usingthe interpretation in [6] as a self linking number of the self intersection. In

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analogy with the encomplexed writhe, the self intersection of a generic realalgebraic surface in RP3 is a real algebraic curve with points of three localcharacters: an intersection of two real sheets, an intersection of two complexconjugate sheets or a Whitney umbrella. Using the local structure around theself intersection, together with a Riemannian metric on RP3 ⊂ CP3, a con-tinuous family of quadratic forms is constructed along the self intersection.Letting MS denote the space of real algebraic mappings from some smoothprojective real algebraic surface S into RP3 two discriminants, σ ,γ ⊂MS, aredefined. The discriminant σ consists of those points in MS such that the corre-sponding parametrized surface in RP3 has topologically unstable singularities.The discriminant γ consists consists of those points in MS such that some pointin the self intersection has an associated quadratic form with one eigenvalueof multiplicity two. We construct an invariant called the fourfold pushoff in-variant, which is defined on points in MS \(σ ∪γ), and show that this invariantis constant on connected components of MS \ (σ ∪ γ). Furthermore, we alsoshow that in the case of the real algebraic surface being an immersed surfacewithout solitary self intersections the Brown invariant coincides with the four-fold pushoff invariant. Counted modulo 8, the fourfold pushoff invariant isshown to be constant on connected components of MS \ γ .

1.3 Contact Geometry and Legendrian ContactHomology

Contact geometry is in many ways the odd dimensional counterpart of sym-plectic geometry. We say that a manifold M is a contact manifold with acontact form α if M is an odd dimensional smooth manifold equipped witha 1-form α such that α ∧ (dα)n is the volume form. Darbouxs theorem tellsus that all contact manifolds locally look alike. The 1-form α can then bedescribed by dz−∑xidyi in local coordinates (x1,y1,x2,y2, ...,xn,yn,z).Thecontact form α defines a distribution of hyperplanes, by simply taking thesubbundle of T M which lies in the kernel of α . We say that a submanifold ofM is Legendrian if it is tangent to this distribution and of maximal dimension,i.e. , at every point its tangent plane lies in the distribution and dim(L) = n.

In Paper III we study Legendrian knots in P×R, where P is a puncturedRiemann surface. Here the symplectic form ω on the Riemann surface isexact, ω = dθ and the contact form on P×R is α = dz− θ , where z is acoordinate along the R-factor. A knot is said to be Legendrian if it is every-where tangent to the contact distribution ξ = ker(α). The Reeb vector fieldR of a contact form α is characterized by dα(R, ·) = 0 and α(R) = 1. In thecase P×R, R = ∂z. Note that the differential of the Lagrangian projectionπ : P×R→ P is an isomorphism when restricted to the contact planes in ξ .Pulling back the complex structure on P to ξ we get a complex structure J

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compatible with dα . Let K be an oriented Legendrian knot. We say that K0and K1 are Legendrian isotopic if there exists a smooth isotopy Kt such thatKt0 is a Legendrian knot for each t0 ∈ [0,1].

Chekanov [1] and Eliashberg [2] showed that there exist formally Legen-drian isotopic knots in R2×R that are not Legendrian isotopic using Legen-drian contact homology. Both proofs used linearized contact homology, a the-ory which was later incorporated in the theoretical framework by Eliashberg,Givental and Hofer in [10] introducing symplectic field theory. Legendriancontact homology associates a differential graded algebra (DGA) to a Legen-drian knot K. The DGA is generated by Reeb chords on K, i.e. flow lines ofR starting and ending on K. The differential is given by a holomorphic curvecount in the symplectization of the contact manifold. The quasi-isomorphismtype of the DGA (in particular its homology) is invariant under Legendrian iso-topy. In [7], Ekholm, Etnyre and Sullivan worked out the details of Legendriancontact homology in the case of a contact manifold of the form P×R, whereP is an exact symplectic manifold of any even dimension 2n. If Λ ⊂ P×Ris Legendrian then π : Λ→ P is a Lagrangian immersion and Reeb chords ofΛ correspond double points of this immersion. In [7], a complex structure onthe contact planes which is pulled back from an almost complex structure onP was used. For such a complex structure, holomorphic disks in P×R withboundary on Λ×R can be described in terms of holomorphic disks in P withboundary on π(Λ), and the DGA of Λ was shown to be invariant under Leg-endrian isotopies up to stable tame isomorphism.In Paper III we describe how to compute the Legendrian contact homologycombinatorially when P is a punctured Riemann surface. Similar situationshave also been studied by other authors, e.g. Ng and Traynor in [3], wherethey give a combinatorial interpretation of contact homology in J1(S1). IfK is a Legendrian knot in P×R then the DGA of K is generated by cross-ings of the knot diagram of K in P, and the differential can be computed bycounting rigid holomorphic disks with boundary on the knot diagram. By theRiemann mapping theorem, such disks correspond to immersed polygons inP with boundary on the knot diagram. In order to construct and work withLegendrian knots in R2×R it is often more convenient to work with knot dia-grams in the front projection: if θ = ydx, then the front projection projects outthe y-coordinate. For generic knots the front diagram is a self transverse im-mersion without vertical tangents away from a finite number of semi-cubicalcusps. The front diagram determines the knot completely and it was shownby Ng in [4] how to recover a Lagrangian diagram from a front diagram andhence how to compute the DGA.In Paper III, we also introduce the notion of a front diagram for Legendrianknots in P×R for P 6= R2, and show how to construct examples of Leg-endrian knots in P×R using this construction. By constructing appropri-ate Legendrian knots and using the Legendrian contact homology we show

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that for any h ∈ H1(P×R) and any positive integer k there exists Legen-drian knots K1, . . . ,Kk realizing the homology class h such that Ki and K jare smoothly Legendrian isotopic, and take the same value for the classicalLegendrian invariants however Ki and K j are not Legendrian isotopic if i 6= j,i, j ∈ {1, . . . ,k}. The proof makes use of knots Kh in classes h 6= 0 which havethe property that the differential of each generator in the associated DGA is 0.

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2. Summary in Swedish (sammanfattning påsvenska)

Denna avhandling studerar knutar och ytor med starkare geometriska struk-turer i förhållande till det klassiska släta fallet. Avhandlingen består av treartiklar. I Artikel I undersöks reellalgebraiska rationella knutar i det reellaprojektiva rummet upp till rigid isotopi och en komplett rigid isotopi klas-sifikation uppnås för sådana knutar av grad högst 5. I Artikel II generalise-ras Browns invariant till generiska parametriserade reellalgebraiska ytor i dettredimensionella projektiva rummet. Artikel III behandlar Legendriska knutari kontaktmångfalden P×R, där P är en punkterad Riemannyta. Det existe-rar en invariant för Legendriska delmångfalder av kontaktmångfalder, kalladkontakthomologi, given av geometriska definitioner. I Artikel III beskrivs hurdenna invariant kan beräknas kombinatoriskt.

2.1 Reellalgebraisk KnutteoriI klassisk knutteori studeras inbäddningar av kompakta 1-mångfalder (cirklar)i det tredimensionella rummet. Den reellalgebraiska motsvarigheten är reellal-gebraiska inbäddningar av reellalgebraiska kurvor i det tredimensionella pro-jektiva rummet RP3. Istället för att betrakta knutar upp till slät isotopi, så be-traktar vi våra reellalgebraiska knutar upp till rigid isotopi. Två reellalgebrais-ka knutar är rigid-isotopa om det existerar en slät isotopi som i varje ögonblickinte bara ger en slät knut, utan också en algebraisk. Detta motsvarar att det ex-isterar en väg mellan dessa två knutar i rummet av alla algebraiska knutar. Dåvarje reellalgebraisk knut kommer utrustad med en grad d, och två reellalge-braiska knutar av olika grader aldrig är rigid-isotopa så får vi en klassificeringupp till rigid isotopi för varje given grad d. I artikel I klassificeras alla rationel-la reellalgebraiska knutar av grad d ≤ 5 upp till rigid isotopi. Då det existerarpar av reellalgebraiska knutar som är slätt isotopa men inte rigid-isotopa så be-hövs ytterligare reellalgebraiska invarianter bortom den klassiska knutteorinssläta invarianter. I Artikel I visas att Viros komplexifierade självlänkningstalfrån [8] är en komplett invariant av rationella reellalgebraiska knutar av gradd ≤ 5, det vill säga, två knutar har samma komplexifierade självlänkningstalom, och endast om, de är rigid-isotopa. Det visas också att detta inte gälleri högre grader. Det komplexifierade självlänkningstalet räknar dubbelpunkterefter en projektion till något generiskt plan med tecken. Till skillnad från den

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klassiska situationen så finns det två typer av dubbelpunkter under en generiskprojektion, de som kommer från reella bitar av knuten (som man också seri det klassiska fallet) och de som kommer från komplexkonjugerade bitar avknuten. Den senare typen av dubbelpunkter kallas solitära. Se Figur 1.1 för ettexempel på när en ickesolitär dubbelpunkt blir solitär.

2.2 Invarianter för Reellalgebraiska YtorI Artikel II introduceras en generalisering av Browns invariant till reellalgebra-iska ytor. I [6] presenteras en tolkning av Browns invariant som ett självlänk-ningstal för självskärningen av en immerserad yta. En generisk immerserad ytahar en självskärningen som i komplementet till ett ändligt antal punkter har enlokal beskrivning som en skärning av två reella plan. Generiska parametrisera-de reellalgebraiska ytor har en självskärning som är en reellalgebraisk kurva. Ikomplementet av ett ändligt antal punkter så kan denna kurva lokalt beskrivassom antingen en skärning mellan två reella plan eller som en skärning mel-lan två komplexkonjugerade plan. Kurvan övergår från den ena situationen tillden andra då den passerar ett Whitney-paraply. En illustration av Whitney-paraplyet finns i Figur 2.1. I Artikel II beskrivs hur den lokala strukturen kringdenna självskärning ger en möjlig generalisering av Browns invariant.

Figur 2.1. Ett Whitney-paraply passeras då en solitär självskärning blir en ickesolitärsjälvskärning.

2.3 Kontaktgeometri och KontakthomologiEn kontaktmångfald M är en slät orienterbar 2n+1-mångfald utrustad med en1−form α sådan att α ∧ (dαn) är volymformen på mångfalden M. Vi säger

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att en delmångfald L ⊂M är Legendrisk om L ⊂ ker(α). Två delmångfalderL och L′ sägs vara Legendriskt isotopa om det existerar en isotopi som tar Ltill L′ sådan att den är Legendrisk vid varje tidpunkt på vägen. I Artikel IIIger vi en kombinatorisk beskrivning av Legendrisk kontakthomologi i P×R,där P är en punkterad Riemannyta. Den kombinatoriska beskrivningen utgårfrån en artikel av Ekholm, Etnyre och Sullivan [7] där kontakthomologi förrum på formen P×R beskrivs geometriskt för mer generella val av P. I dettredimensionella fallet så kan dessa definitioner reduceras till kombinatorik.

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3. Acknowledgements

First I would like to thank my advisor Tobias Ekholm for inspiration and forsharing with me his deep knowledge of mathematics. I have benefitted greatlyfrom your advice, both concerning my research in particular and concerningmathematics and being a mathematician in general.I would also like to thank Oleg Viro who has been my teacher during mostof my undergraduate studies and my advisor during the first years of my PhDstudies. Your vision of mathematics and your inspiring lectures and discus-sions have had a great influence on me, both as a mathematician and as ateacher.I am also very grateful to my assistant advisor, Ryszard Rubinsztein. You havealways had the time to answer my (sometimes strange) questions concerningmany different topics in mathematics.I would like to thank the Mittag Leffler institute and especially the organiz-ers: Alicia Dickenstein, Sandra Di Rocco, Ragni Piene, Kristian Ranestad andBernd Sturmfels for hospitality and for a very interesting algebraic geometrysemester.During my time in Uppsala, both during and before my PhD studies, I havehad many excellent teachers. I would especially like to thank the following:Evgeny Schepin and Anders Vretblad for giving me an inspiring and chal-lenging first semester, Karl-Heinz Fieseler for his many excellent courses andexercises which I have spent countless hours learning from and Gunnar Bergfor many interesting discussions.Many thanks goes out to my fellow PhD students, especially the ”fredagsfika“-group for interesting conversations and pleasant times. There are three per-sons I would like to thank in particular: Anders Södergren for the many hoursspent discussing mathematics and teaching, Cecilia Holmgren, for being agreat friend (and now co-author) and for our many interesting conversations,both mathematical and non-mathematical, and Isac Hedén for the many latenight mathematics discussions and for the pleasant times working together onour Algebra II course.Among my friends outside the department, I would especially like to thankValentina Chapovalova, Eric Fridén, Jon-Erik Karlsson, Erik Strandberg, Lin-nea Talltjärn and Per Wimelius.Last, but certainly not least, I would like to thank my family.

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References

[1] Y. Chekanov. New invariants of legendrian knots. Progr. Math., 202:525–534,2001.

[2] Y. Eliashberg. Invariants in contact topology. Proceedings of the InternationalCongress of Mathematicians, II:327–338, 1998.

[3] Lenhard; L. Ng and L. Traynor. Legendrian solid-torus links. J. SymplecticGeom., 2:411–443, 2004.

[4] L. Ng. Computable legendrian invariants. Topology, 42(1):55–82, 2003.[5] S. Orevkov. Classification of algebraic links in RP3 of degree 5 and 6.

Presented at the conference Perspectives in analysis, geometry and topology,Stockholm, 2008.

[6] P. Melvin R .Kirby. Local surgery formulas for quantum invariants and the arfinvariant. Geom. Topol. Monogr. 7, 7, 2004.

[7] T. Ekholm J. Etnyre and M. Sullivan. Legendrian contact homology in P×R.Transactions of the American Mathematical Society, 359:3301–3335, 2007.

[8] O. Viro. Encomplexing the writhe. In Topology, ergodic theory, real algebraicgeometry, volume 202 of Amer. Math. Soc. Transl. Ser. 2, pages 241–256. Amer.Math. Soc., Providence, 2001.

[9] O. Viro. Whitney number of closed real algebraic affine curve of type I.Moscow Mathematical Journal 6:1, 2006.

[10] Ya. Eliashberg A. Givental and H. Hofer. Introduction to symplectic fieldtheory. Geom. Funct. Anal., pages 560–673, 2000. Special Volume, Part II.

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