On knot Floer homology and lens space surgeries

In an earlier paper, we used the absolute grading on Heegaard Floer homology HF⁺ to give restrictions on knots in S³ which admit lens space surgeries. The aim of the present article is to exhibit stronger restrictions on such knots, arising from knot Floer homology. One consequence is that the non-zero coefficients of the Alexander polynomial of such a knot are ±1. This information can in turn be used to prove that certain lens spaces are not obtained as integral surgeries on knots. In fact, combining our results with constructions of Berge, we classify lens spaces L(p,q) which arise as integral surgeries on knots in S³ with |p|?1500. Other applications include bounds on the four-ball genera of knots admitting lens space surgeries (which are sharp for Berge's knots), and a constraint on three-manifolds obtained as integer surgeries on alternating knots, which is closely to related to a theorem of Delman and Roberts.     

Absolutely graded Floer homologies and intersection forms for four-manifolds with boundary

In Ozsváth and Szabó (Holomorphic triangles and invariants for smooth four-manifolds, math. SG/0110169, 2001), we introduced absolute gradings on the three-manifold invariants developed in Ozsváth and Szabó (Holomorphic disks and topological invariants for closed three-manifolds, math.SG/0101206, Ann. of Math. (2001), to appear). Coupled with the surgery long exact sequences, we obtain a number of three- and four-dimensional applications of this absolute grading including strengthenings of the “complexity bounds” derived in Ozsváth and Szabó (Holomorphic disks and three-manifold invariants: properties and applications, math.SG/0105202, Ann. of Math. (2001), to appear), restrictions on knots whose surgeries give rise to lens spaces, and calculations of HF⁺ for a variety of three-manifolds. Moreover, we show how the structure of HF⁺ constrains the exoticness of definite intersection forms for smooth four-manifolds which bound a given three-manifold. In addition to these new applications, the techniques also provide alternate proofs of Donaldson's diagonalizability theorem and the Thom conjecture for .     

Vassiliev invariants of Legendrian, transverse, and framed knots in contact three-manifolds

We show that for a large class of contact three-manifolds the groups of Vassiliev invariants of Legendrian and of framed knots are canonically isomorphic. As a corollary, we obtain that the group of finite order Arnold's J⁺-type invariants of wave fronts on a surface F is isomorphic to the group of Vassiliev invariants of framed knots in the spherical cotangent bundle ST∗F of F.On the other hand, we construct the first examples of contact manifolds for which Vassiliev invariants of Legendrian knots can distinguish Legendrian knots that realize isotopic framed knots and are homotopic as Legendrian immersions.     

On the Heegaard Floer homology of a surface times a circle

We make a detailed study of the Heegaard Floer homology of the product of a closed surface Σg of genus g with S¹. We determine HF⁺(Σg×S¹,s;C) completely in the case c₁(s)=0, which for g?3 was previously unknown. We show that in this case HF^∞ is closely related to the cohomology of the total space of a certain circle bundle over the Jacobian torus of Σg, and furthermore that HF⁺(Σg×S¹,s;Z) contains nontrivial 2-torsion whenever g?3 and c₁(s)=0. This is the first example known to the authors of torsion in Z-coefficient Heegaard Floer homology. Our methods also give new information on the action of H₁(Σg×S¹) on HF⁺(Σg×S¹,s) when c₁(s) is nonzero.     

On three-manifolds dominated by circle bundles

We determine which three-manifolds are dominated by products. The result is that a closed, oriented, connected three-manifold is dominated by a product if and only if it is finitely covered either by a product or by a connected sum of copies of S ² × S ¹. This characterization can also be formulated in terms of Thurston geometries, or in terms of purely algebraic properties of the fundamental group. We also determine which three-manifolds are dominated by non-trivial circle bundles, and which three-manifold groups are presentable by products.     

Embedding infinite cyclic covers of knot spaces into 3-space

We say a knot k in the 3-sphere S³ has PropertyIE if the infinite cyclic cover of the knot exterior embeds into S³. Clearly all fibred knots have Property IE.There are infinitely many non-fibred knots with Property IE and infinitely many non-fibred knots without property IE. Both kinds of examples are established here for the first time. Indeed we show that if a genus 1 non-fibred knot has Property IE, then its Alexander polynomial Δ_k(t) must be either 1 or 2t²−5t+2, and we give two infinite families of non-fibred genus 1 knots with Property IE and having Δ_k(t)=1 and 2t²−5t+2 respectively.Hence among genus 1 non-fibred knots, no alternating knot has Property IE, and there is only one knot with Property IE up to ten crossings.We also give an obstruction to embedding infinite cyclic covers of a compact 3-manifold into any compact 3-manifold.     

Topology of Maximally Writhed Real Algebraic Knots

Oleg Viro introduced an invariant of rigid isotopy for real algebraic knots in ??³ which can be viewed as a first order Vassiliev invariant. In this paper we look at real algebraic knots of degree d with the maximal possible value of this invariant. We show that for a given d all such knots are topologically isotopic and explicitly identify their knot type.     

Les invariants rationnels de type fini ne distinguent pas les nœuds dans SS2×SS1

We introduce geometric sequences of knots and establish the following criterion: if v is a rational invariant of degree ≤m in the sense of Vassiliev, then v is a polynomial of degree ≤m on every geometric sequence of knots. The torsion in the braid group over the sphere induces torsion at the level of Vassiliev invariants: we construct knots in SS²×SS¹ which cannot be distinguished by rational invariants of finite type. They can, however, be distinguished by invariants of finite type with values in a finite abelian group.     

Legendrian Knots in Overtwisted Contact Structures on S3

We study the problem of classifying Legendrian knots in overtwisted contact structures on S ³. The question is whether topologically isotopic Legendrian knots have to be Legendrian isotopic if they have equal values of the well-known invariants rot and tb. We give positive answer in the case that there is an overtwisted disc intersecting none of the knots and we construct an example of a knot intersecting each overtwisted disc (this provides a counterexample to the conjecture of Eliashberg). Our proof needs some results on the structure of the group of contactomorphisms of S ³. We divide the subgroup Cont₊(S ³, ) of coorientation-preserving contactomorphisms for an overtwisted contact distribution into two classes.     

Surgery obstructions from Khovanov homology

For a 3-manifold with torus boundary admitting an appropriate involution, we show that Khovanov homology provides obstructions to certain exceptional Dehn fillings. For example, given a strongly invertible knot in S ³, we give obstructions to lens space surgeries, as well as obstructions to surgeries with finite fundamental group. These obstructions are based on homological width in Khovanov homology, and in the case of finite fundamental group depend on a calculation of the homological width for a family of Montesinos links.     

The isocohomological property, higher Dehn functions, and relatively hyperbolic groups

The property that the polynomial cohomology with coefficients of a finitely generated discrete group is canonically isomorphic to the group cohomology is called the (weak) isocohomological property for the group. In the case when a group is of type HF^∞, i.e. that has a classifying space with the homotopy type of a polyhedral complex with finitely many cells in each dimension, we show that the isocohomological property is geometric and is equivalent to the property that the universal cover of the classifying space has polynomially bounded higher Dehn functions. If a group is hyperbolic relative to a collection of subgroups, each of which is polynomially combable, respectively HF^∞ and isocohomological, then we show that the group itself has these respective properties. Combining with the results of Connes-Moscovici and Dru?u-Sapir we conclude that a group satisfies the strong Novikov conjecture if it is hyperbolic relative to subgroups which are of property RD, of type HF^∞ and isocohomological.     

Necessary and Sufficient Condition for Lorenz Knots to be Closed Under Satellite Construction

Lorenz knots and links are the periodic orbits of a certain system of differnetial equations in R³. In this work, we give the necessary and sufficient condition for Lorenz knots to be closed under satellite construction. We prove precisely that the only possible satellites of a Lorenz knot are parallels with possible twists.     

Thin presentation of knots in lens spaces and RP3-conjecture

This Note concerns knots in a lens space L that produce S³ by Dehn surgery. We introduce the thin presentation of knots in L, with respect to a standard spine. We prove that among such knots, those having a thin presentation with only maxima, are 0-bridge or 1-bridge braids in L. In the case $\text{L=}\text{R}\text{P}{}^3$, we deduce that minimally braided knots in $\text{R}\text{P}{}^3$ cannot yield S³ by Dehn surgery. To cite this article: A. Deruelle, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

High distance Heegaard splittings via fat train tracks

We define fat train tracks and use them to give a combinatorial criterion for the Hempel distance of Heegaard splittings for closed orientable 3-manifolds. We apply this criterion to 3-manifolds obtained from surgery on knots in S³.     

Almost contact Riemannian three-manifolds with Reeb flow symmetry

In this paper we give a complete classification of simply connected homogeneous almost α-Kenmotsu three-manifolds M whose Ricci operator is invariant along the Reeb flow. We get this classification by using the Gaussian and the extrinsic curvature associated with the canonical foliation of M.     

Seifert fibered surgeries with distinct primitive/Seifert positions

We call a pair (K,m) of a knot K in the 3-sphere S³ and an integer m a Seifert fibered surgery if m-surgery on K yields a Seifert fiber space. For most known Seifert fibered surgeries (K,m), K can be embedded in a genus 2 Heegaard surface of S³ in a primitive/Seifert position, the concept introduced by Dean as a natural extension of primitive/primitive position defined by Berge. Recently Guntel has given an infinite family of Seifert fibered surgeries each of which has distinct primitive/Seifert positions. In this paper we give yet other infinite families of Seifert fibered surgeries with distinct primitive/Seifert positions from a different point of view.     

Local structure of ideal shapes of knots

Relatively extremal knots are the relative minima of the ropelength functional in the C¹ topology. They are the relative maxima of the thickness (normal injectivity radius) functional on the set of curves of fixed length, and they include the ideal knots. We prove that a C1,1 relatively extremal knot in Rn either has constant maximal (generalized) curvature, or its thickness is equal to half of the double critical self distance. This local result also applies to the links. Our main approach is to show that the shortest curves with bounded curvature and C¹ boundary conditions in Rn contain CLC (circle-line-circle) curves, if they do not have constant maximal curvature.