Knots and Surfaces in Real Algebraic and Contact Geometry

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UPPSALA DISSERTATIONS IN MATHEMATICS

72

Knots and Surfaces in Real

Algebraic and Contact Geometry

Johan Björklund

Department of Mathematics

Uppsala University

UPPSALA 2011

Department of Mathematics

Uppsala University

UPPSALA 2011

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List of papers

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

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

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

Submitted

Preprint available at arxiv.org/abs/1108.1566

III Björklund, J. Legendrian Contact Homology in the Product of a Punctured Riemann Surface and the Real Line

Submitted

Preprint available at arxiv.org/abs/1108.1568

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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 algebraic rational knots in RP3are 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 an invariant for smooth topology, the so-called Brown invariant, is generalized to generic parametrized real algebraic surfaces in RP3. Paper III concerns Leg-endrian knots in the contact manifold P × R, where P is a punctured Riemann surface. To distinguish Legendrian submanifolds of contact manifolds there exists an invariant called contact homology. This invariant is defined using a geometric description, which comes from symplectic field theory. In Paper III we describe how to calculate this invariant in a combinatorial manner in this low dimensional setting.

1.1 Real Algebraic Knot Theory

The real algebraic counterpart to 1-manifolds are real algebraic curves, i.e., al-gebraic curves defined by polynomials with real coefficients. Just as a smooth knot in some space X is an embedding of S1 into X , a real algebraic knot is an injective real algebraic map of some real algebraic curve C into some real algebraic 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. Two real algebraic knots are then said to be rigidly isotopic if there exists a path in the space of real algebraic knots connecting them. In Paper I we consider the problem of classifying real algebraic knots in projective space up to rigid iso-topy. There are several kinds of algebraic curves, classified by genus. There has been work done by Mikhalkin and Orevkov [5] classifying real algebraic knots 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_{. Since}

the 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 rigidly isotopic. 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 smooth knot theory are rigid isotopy invariants since any rigid isotopy is in particular an isotopy. Finding invariants to distinguish knots up to smooth isotopy has been 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 point to 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 Khovanov homology, a categorification of the Jones polynomial. However, there exist real algebraic knots of the same degree which are smoothly isotopic but not rigidly isotopic, thus necessitating additional invariants beyond those used in smooth knot theory. The only known invariant (which does not come from smooth isotopy or the degree) is the encomplexed writhe discovered by Viro [8]. This invariant uses the natural complexification CK of a real algebraic knot K ⊂ RP3, by simply extending the appropriate map f : RP1→ RP3 _to

a map C f : CP1→ CP3_{. Given a generic projection to a plane in RP}3 _we

get some finite number of double points. In contrast to smooth knot theory, where each such double point arises from two transversal pieces of the knot intersecting, 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 solitary double point. By assigning appropriate values ±1 to these double points and then summing them, an invariant is obtained. In Paper I it was proved that two real algebraic rational knots of degree d ≤ 5 are rigidly isotopic if, and only if, the two knots have coinciding values of their encomplexed writhe. It was also shown that this does not hold true for higher degrees by the construction of a counterexample in degree d = 6. Modeling RP3as a ball with antipodal points identified, we illustrate all real algebraic knots of degree d ≤ 5 up to smooth isotopy and mirror image in Figure 1.2.

1.2 Invariants in Real Algebraic Geometry

Following the philosophy of Viro in [8] we consider an invariant in the real algebraic world to be encomplexed if it is a natural extension of an invariant from the world of smooth topology. The encomplexed writhe was one

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Figure 1.2. All real algebraic rational knots of degree d ≤ 5, up to smooth isotopy and 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. Another invariant that survives from the smooth topological world is the Whitney index for curves in the plane. We recall that an immersion is a map with injective differential. 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 Whitney index. The Whitney index can be calculated from the self intersections in the case 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 study a similar situation concerning the space of generically immersed oriented sur-faces in R3. The Brown invariant is an invariant up to regular homotopy of immersed 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 and Melvin [6]. In their article they express the Brown invariant by constructing an auxiliary curve called the “pushoff“ on the boundary of a tubular neigh-bourhood of the self intersection together with a natural projection to the self intersection 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 to give the Brown invariant.

In Paper II we gave a definition of an encomplexed Brown invariant, using the 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 real algebraic surface in RP3 is a real algebraic curve with points of three local characters: an intersection of two real sheets, an intersection of two complex conjugate sheets or a Whitney umbrella. Using the local structure around the self 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 smooth

projective real algebraic surface S into RP3two discriminants, σ , γ ⊂ MS, are

defined. The discriminant σ consists of those points in MSsuch that the

corre-sponding parametrized surface in RP3has topologically unstable singularities. The discriminant γ consists consists of those points in MSsuch that some point

in the self intersection has an associated quadratic form with one eigenvalue of 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 invariant

is constant on connected components of MS\ (σ ∪ γ). Furthermore, we also

show that in the case of the real algebraic surface being an immersed surface without solitary self intersections the Brown invariant coincides with the four-fold pushoff invariant. Counted modulo 8, the fourfour-fold pushoff invariant is shown to be constant on connected components of MS\ γ.

1.3 Contact Geometry and Legendrian Contact

Homology

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 a contact form α if M is an odd dimensional smooth manifold equipped with a 1-form α such that α ∧ (dα)n is the volume form. Darbouxs theorem tells us that all contact manifolds locally look alike. The 1-form α can then be described by dz − ∑ xidyi in local coordinates (x1, y1, x2, y2, ..., xn, yn, z).The

contact form α defines a distribution of hyperplanes, by simply taking the subbundle of T M which lies in the kernel of α. We say that a submanifold of Mis 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 punctured Riemann surface. Here the symplectic form ω on the Riemann surface is exact, ω = dθ and the contact form on P × R is α = dz − θ , where z is a coordinate 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 field Rof a contact form α is characterized by dα(R, ·) = 0 and α(R) = 1. In the case P × R, R = ∂z. Note that the differential of the Lagrangian projection

π : 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 K0

and K1 are Legendrian isotopic if there exists a smooth isotopy Kt such that

Kt₀ 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. Legendrian contact 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 of Rstarting and ending on K. The differential is given by a holomorphic curve count in the symplectization of the contact manifold. The quasi-isomorphism type of the DGA (in particular its homology) is invariant under Legendrian iso-topy. In [7], Ekholm, Etnyre and Sullivan worked out the details of Legendrian contact homology in the case of a contact manifold of the form P × R, where P_{is an exact symplectic manifold of any even dimension 2n. If Λ ⊂ P × R} is Legendrian then π : Λ → P is a Lagrangian immersion and Reeb chords of Λ correspond double points of this immersion. In [7], a complex structure on the contact planes which is pulled back from an almost complex structure on Pwas used. For such a complex structure, holomorphic disks in P × R with boundary on Λ × R can be described in terms of holomorphic disks in P with boundary 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 homology combinatorially when P is a punctured Riemann surface. Similar situations have also been studied by other authors, e.g. Ng and Traynor in [3], where they give a combinatorial interpretation of contact homology in J1(S1). If K _{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 by counting rigid holomorphic disks with boundary on the knot diagram. By the Riemann mapping theorem, such disks correspond to immersed polygons in Pwith boundary on the knot diagram. In order to construct and work with Legendrian knots in R2× R it is often more convenient to work with knot dia-grams in the front projection: if θ = y dx, then the front projection projects out the y-coordinate. For generic knots the front diagram is a self transverse im-mersion without vertical tangents away from a finite number of semi-cubical cusps. The front diagram determines the knot completely and it was shown by Ng in [4] how to recover a Lagrangian diagram from a front diagram and hence how to compute the DGA.

In Paper III, we also introduce the notion of a front diagram for Legendrian knots 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 Kj

are smoothly Legendrian isotopic, and take the same value for the classical Legendrian invariants however Kiand Kjare not Legendrian isotopic if i 6= j,

i, j ∈ {1, . . . , k}. The proof makes use of knots Khin classes h 6= 0 which have

the property that the differential of each generator in the associated DGA is 0.

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2. Summary in Swedish (sammanfattning på

svenska)

Denna avhandling studerar knutar och ytor med starkare geometriska struk-turer i förhållande till det klassiska släta fallet. Avhandlingen består av tre artiklar. I Artikel I undersöks reellalgebraiska rationella knutar i det reella projektiva 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 det tredimensionella projektiva rummet. Artikel III behandlar Legendriska knutar i kontaktmångfalden P × R, där P är en punkterad Riemannyta. Det existe-rar en invariant för Legendriska delmångfalder av kontaktmångfalder, kallad kontakthomologi, given av geometriska definitioner. I Artikel III beskrivs hur denna invariant kan beräknas kombinatoriskt.

2.1 Reellalgebraisk Knutteori

I klassisk knutteori studeras inbäddningar av kompakta 1-mångfalder (cirklar) i det tredimensionella rummet. Den reellalgebraiska motsvarigheten är reellal-gebraiska inbäddningar av reellalreellal-gebraiska 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 ögonblick inte 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. Då varje reellalgebraisk knut kommer utrustad med en grad d, och två reellalge-braiska knutar av olika grader aldrig är rigid-isotopa så får vi en klassificering upp till rigid isotopi för varje given grad d. I artikel I klassificeras alla rationel-la reelrationel-lalgebraiska knutar av grad d ≤ 5 upp till rigid isotopi. Då det existerar par av reellalgebraiska knutar som är slätt isotopa men inte rigid-isotopa så be-hövs ytterligare reellalgebraiska invarianter bortom den klassiska knutteorins släta invarianter. I Artikel I visas att Viros komplexifierade självlänkningstal från [8] är en komplett invariant av rationella reellalgebraiska knutar av grad d≤ 5, det vill säga, två knutar har samma komplexifierade självlänkningstal om, och endast om, de är rigid-isotopa. Det visas också att detta inte gäller i högre grader. Det komplexifierade självlänkningstalet räknar dubbelpunkter efter 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 generisk projektion, de som kommer från reella bitar av knuten (som man också ser i det klassiska fallet) och de som kommer från komplexkonjugerade bitar av knuten. Den senare typen av dubbelpunkter kallas solitära. Se Figur 1.1 för ett exempel på när en ickesolitär dubbelpunkt blir solitär.

2.2 Invarianter för Reellalgebraiska Ytor

I 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 yta har en självskärningen som i komplementet till ett ändligt antal punkter har en lokal beskrivning som en skärning av två reella plan. Generiska parametrisera-de reellalgebraiska ytor har en självskärning som är en reellalgebraisk kurva. I komplementet av ett ändligt antal punkter så kan denna kurva lokalt beskrivas som antingen en skärning mellan två reella plan eller som en skärning mel-lan två komplexkonjugerade pmel-lan. Kurvan övergår från den ena situationen till den andra då den passerar ett paraply. En illustration av Whitney-paraplyet finns i Figur 2.1. I Artikel II beskrivs hur den lokala strukturen kring denna självskärning ger en möjlig generalisering av Browns invariant.

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

2.3 Kontaktgeometri och Kontakthomologi

En kontaktmångfald M är en slät orienterbar 2n + 1-mångfald utrustad med en 1−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 delmångfalder Loch L0 sägs vara Legendriskt isotopa om det existerar en isotopi som tar L till L0 sådan att den är Legendrisk vid varje tidpunkt på vägen. I Artikel III ger vi en kombinatorisk beskrivning av Legendrisk kontakthomologi i P × R, där P är en punkterad Riemannyta. Den kombinatoriska beskrivningen utgår från en artikel av Ekholm, Etnyre och Sullivan [7] där kontakthomologi för rum på formen P × R beskrivs geometriskt för mer generella val av P. I det tredimensionella 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 for sharing with me his deep knowledge of mathematics. I have benefitted greatly from your advice, both concerning my research in particular and concerning mathematics and being a mathematician in general.

I would also like to thank Oleg Viro who has been my teacher during most of my undergraduate studies and my advisor during the first years of my PhD studies. Your vision of mathematics and your inspiring lectures and discus-sions have had a great influence on me, both as a mathematician and as a teacher.

I am also very grateful to my assistant advisor, Ryszard Rubinsztein. You have always had the time to answer my (sometimes strange) questions concerning many 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 and Bernd Sturmfels for hospitality and for a very interesting algebraic geometry semester.

During my time in Uppsala, both during and before my PhD studies, I have had 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 and exercises which I have spent countless hours learning from and Gunnar Berg for 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 hours spent discussing mathematics and teaching, Cecilia Holmgren, for being a great friend (and now co-author) and for our many interesting conversations, both mathematical and non-mathematical, and Isac Hedén for the many late night mathematics discussions and for the pleasant times working together on our Algebra II course.

Among my friends outside the department, I would especially like to thank Valentina 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 International Congress of Mathematicians, II:327–338, 1998.

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

[4] L. Ng. Computable legendrian invariants. Topology, 42(1):55–82, 2003. [5] S. Orevkov. Classification of algebraic links in RP3of 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 arf invariant. 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 algebraic

geometry, 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 field theory. Geom. Funct. Anal., pages 560–673, 2000. Special Volume, Part II.

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Knots and Surfaces in Real Algebraic and Contact Geometry

UPPSALA DISSERTATIONS IN MATHEMATICS

72