We show there exists a linear function w: N->N with the following property. Let K be a hyperbolic knot in a hyperbolic 3-manifold M admitting a non-longitudinal S^3 surgery. If K is put into thin position with respect to a strongly irreducible, genus g Heegaard splitting of M then K intersects a thick level at most 2w(g) times. Typically, this shows that the bridge number of K with respect to this Heegaard splitting is at most w(g), and the tunnel number of K is at most w(g) + g-1.
We give a parity condition of a Heegaard diagram to show that it is unstabilized. This improves the result of [5]. As an application, we construct unstabilized Heegaard splittings by Dehn twists on any given Heegaard splitting.
We use an algorithm by Ozsvath and Szabo to find closed formulae for the ranks of the hat version of the Heegaard Floer homology groups for non-zero Dehn surgeries on knots in the 3-sphere. As applications we provide new bounds on the number of distinct ranks of the Heegaard Floer groups a Dehn surgery can have. These in turn give a new lower bound on the rational Dehn surgery genus of a rational homology 3-sphere. We also provide novel obstructions for a knot to be a potential counterexample to the Cabling Conjecture.
We expect manifolds obtained by Dehn filling to inherit properties from the knot manifold. To what extent does that hold true for the Heegaard structure? We study four changes to the Heegaard structure that may occur after filling: (1) Heegaard genus decreases, (2) a new Heegaard surface is created, (3) a non-stabilized Heegaard surface destabilizes, and (4) two or more non-isotopic Heegaard surfaces become isotopic. We survey general results that give quite satisfactory restrictions to phenomena (1) and (2) and, in a parallel thread, give a complete classification of when all four phenomena o…
In 2001, J. Hempel proved the existence of Heegaard splittings of arbitrarily high distance by using a high power of a pseudo-Anosov map as the gluing map between two handlebodies. We show that lower bounds on distance can also be obtained when using a high power of a suitably chosen Dehn twist. In certain cases, we can then determine the exact distance of the resulting splitting. These results can be seen as a natural extension of work by A. Casson and C. Gordon in 1987 regarding strongly irreducible Heegaard splittings.
We use Heegaard splittings to give a criterion for a tunnel number one knot manifold to be non-fibered and to have large cyclic covers. We also show that such a knot manifold (satisfying the criterion) admits infinitely many virtually Haken Dehn fillings. Using a computer, we apply this criterion to the 2 generator, non-fibered knot manifolds in the cusped Snappea census. For each such manifold M, we compute a number c(M), such that, for any n>c(M), the n-fold cyclic cover of M is large.
Let K' be a hyperbolic knot in S^3 and suppose that some Dehn surgery on K' with distance at least 3 from the meridian yields a 3-manifold M of Heegaard genus 2. We show that if M does not contain an embedded Dyck's surface (the closed non-orientable surface of Euler characteristic -1), then the knot dual to the surgery is either 0-bridge or 1-bridge with respect to a genus 2 Heegaard splitting of M. In the case M does contain an embedded Dyck's surface, we obtain similar results. As a corollary, if M does not contain an incompressible genus 2 surface, then the tunnel number of K' is at most 2…
We provide a new obstruction for a rational homology 3-sphere to arise by Dehn surgery on a given knot in the 3-sphere. The obstruction takes the form of an inequality involving the genus of the knot, the surgery coefficient, and a count of L-structures on the 3-manifold, that is spin-c structures with the simplest possible associated Heegaard Floer group. Applications include an obstruction for two framed knots to yield the same 3-manifold, an obstruction that is particularly effective when working with families of framed knots. We introduce the rational and integral Dehn surgery genera for a…
An L-space is a rational homology 3-sphere with minimal Heegaard Floer homology. We give the first examples of hyperbolic L-spaces with no symmetries. In particular, unlike all previously known L-spaces, these manifolds are not double branched covers of links in S^3. We prove the existence of infinitely many such examples (in several distinct families) using a mix of hyperbolic geometry, Floer theory, and verified computer calculations. Of independent interest is our technique for using interval arithmetic to certify symmetry groups and non-existence of isometries of cusped hyperbolic 3-manifo…
We use sutured manifold theory, essential laminations and essential branched surfaces to establish the upper bounds of distances between certain types of nonsimple Dehn surgery slopes. This is the revised version of an earlier preprint {\it Dehn surgery and simple manifolds.}
We write down an explicit formula for the $+$ version of the Heegaard Floer homology (as an absolutely graded vector space over an arbitrary field) of the results of Dehn surgery on a knot $K$ in $S^3$ in terms of homological data derived from $CFK^{\infty}(K)$. This allows us to prove some results about Dehn surgery on knots in $S^3$. In particular, we show that for a fixed manifold there are only finitely many alternating knots that can produce it by surgery. This is an improvement on a recent result by Lackenby and Purcell. We also derive a lower bound on the genus of knots depending on the…
In this paper we extend Thurston's hyperbolic Dehn surgery theorem to a class of geometrically infinite hyperbolic 3-manifolds. As an application we prove a modest density theorem for Kleinian groups. We also discuss hyperbolic Dehn surgery on geometrically finite hypebolic cone-manifolds.
Using Taubes' periodic ends theorem, Auckly gave examples of toroidal and hyperbolic irreducible integer homology spheres which are not surgery on a knot in the three-sphere. We give an obstruction to a homology sphere being surgery on a knot coming from Heegaard Floer homology. This is used to construct infinitely many small Seifert fibered examples.
We prove a surgery formula for the renormalized Euler characteristic of Ozsvath and Szabo. Equality between this Euler cahracteristic and the Seiberg-Witten invariant follows for rational homology three-spheres.
A consequence of the Cabling Conjecture of Gonzalez-Acuña and Short is that Dehn surgery on a knot in $S^3$ cannot produce a manifold with more than two connected summands. In the event that some Dehn surgery produces a manifold with three or more connected summands, then the surgery parameter is bounded in terms of the bridge number by a result of Sayari. Here this bound is sharpened, providing further evidence in favour of the Cabling Conjecture.
A Dehn surgery on a knot $K$ in $S^3$ is exceptional if it produces a reducible, toroidal or Seifert fibred manifold. It is known that a large arborescent knot admits no such surgery unless it is a type II arborescent knot. The main theorem of this paper shows that up to isotopy there are exactly three large arborescent knots admitting exceptional surgery, each of which admits exactly one exceptional surgery, producing a toroidal manifold.
For a compact connected 3-submanifold with connected boundary in the 3-sphere, we relate the existence of a Seifert surface system for a surface with a Dehn surgery along a null-homologous link. As its corollary, we obtain a refinement of the Fox's re-embedding theorem.
This article is solicited by C.\ Adams for a special issue of {\it Chaos, Solitons and Fractals\/} devoted to knot theory and its applications. We present some recent results about Dehn surgeries on arborescent knots and links.
Let K be a fibered knot in the 3-sphere. We show that if the monodromy of K is sufficiently complicated, then Dehn surgery on K cannot yield a lens space. Work of Yi Ni shows that if K has a lens space surgery then it is fibered. Combining this with our result we see that if K has a lens space surgery then it is fibered and the monodromy is relatively simple.
Let M be a closed 3-manifold obtained by Dehn surgery along the figure-eight knot. We give a formula of the Reisdemeisiter torsion of M for any irreducible SL(2;C)-representation. It is described as a rational expression of the trace of the image of the meridian.
Berge in [1] defined doubly primitive knots, which yield lens spaces by Dehn surgery. At the same paper he listed the knots into several types. In this paper we will prove the list is complete when $τ>1$. The invariant $τ$ is a quantity with regard to lens space surgery, which is defined in this paper. Furthermore at the same time we will also prove that Table~6 in [8] is complete as Poincaré homology sphere surgery when $τ>1$.
We provide related Dehn surgery descriptions for rational homology spheres and a class of their regular finite cyclic covering spaces. As an application, we use the surgery descriptions to relate the Casson invariants of the covering spaces to that of the base space. Finally, we show that this places restrictions on the number of finite and cyclic Dehn fillings of the knot complements in the covering spaces beyone those imposed by Culler-Gordon-Luecke-Shalen and Boyer-Zhang.
We prove an explicit, quantitative criterion that ensures the Heegaard surfaces in Dehn fillings behave "as expected." Given a cusped hyperbolic manifold X, and a Dehn filling whose meridian and longitude curves are longer than 2pi(2g-1), we show that every genus g Heegaard splitting of the filled manifold is isotopic to a splitting of the original manifold X. The analogous statement holds for fillings of multiple boundary tori. This gives an effective version of a theorem of Moriah-Rubinstein and Rieck-Sedgwick.
The twisted face-pairing construction of our earlier papers gives an efficient way of generating, mechanically and with little effort, myriads of relatively simple face-pairing descriptions of interesting closed 3-manifolds. The corresponding description in terms of surgery, or Dehn-filling, reveals the twist construction as a carefully organized surgery on a link. In this paper, we work out the relationship between the twisted face-pairing description of closed 3-manifolds and the more common descriptions by surgery and Heegaard diagrams. We show that all Heegaard diagrams have a natural deco…
We calculate limits of the colored Jones polynomials of the figure-eight knot and conclude that in most cases they determine the volumes and the Chern--Simons invariants of the three-manifolds obtained by Dehn surgeries along it.
When can surgery on a null-homologous knot K in a rational homology sphere produce a non-separating sphere? We use Heegaard Floer homology to give sufficient conditions for K to be unknotted. We also discuss some applications to homology cobordism, concordance, and Mazur manifolds.
Let K be a non-trivial knot in the 3-sphere and let Y(r) be the 3-manifold obtained by surgery on K with surgery-coefficient a rational number r. We show that there is a homomorphism from the fundamental group of Y(r) to SU(2) with non-cyclic image if r is less than or equal to 2.
We prove that there are exactly $6$ Nil Seifert fibred spaces which can be obtained by Dehn surgeries on non-trefoil knots in $S^3$, with $\{60, 144, 156, 288, 300\}$ as the exact set of all such surgery slopes up to taking the mirror images of the knots. We conjecture that there are exactly $4$ specific hyperbolic knots in $S^3$ which admit Nil Seifert fibred surgery. We also give some more general results and a more general conjecture concerning Seifert fibred surgeries on hyperbolic knots in $S^3$.
In this paper, we define the primitive/Seifert-fibered property for a knot in S^3. If satisfied, the property ensures that the knot has a Dehn surgery that yields a small Seifert-fibered space (i.e. base S^2 and three or fewer critical fibers). Next we describe the twisted torus knots, which provide an abundance of examples of primitive/Seifert-fibered knots. By analyzing the twisted torus knots, we prove that nearly all possible triples of multiplicities of the critical fibers arise via Dehn surgery on primitive/Seifert-fibered knots.
We show that there exist infinitely many pairs of distinct knots in the 3-sphere such that each pair can yield homeomorphic lens spaces by the same Dehn surgery. Moreover, each knot of the pair can be chosen to be a torus knot, a satellite knot or a hyperbolic knot, except that both cannot be satellite knots simultaneously. This exception is shown to be unavoidable by the classical theory of binary quadratic forms.
Let K be a (2p,q)-torus knot and M_n is a 3-manifold obtained by 1/n-Dehn surgery along K. We consider a polynomial whose zeros are the inverses of the Reideimeister torsion of M_n for SL(2;C)-irreducible representations. Johnson gave a formula for the case of the (2,3)-torus knot under some modification and normalization. We generalize this formula by using Tchebychev polynomials.
We use the combinatorial techniques of graphs of intersection to study reducible Dehn surgeries on knots in the three-sphere. In particular, in the event that a reducible surgery on a knot K in the three-sphere of slope r produces a manifold with more than two connected summands, we show that r is bounded in absolute value by the bridge number of K. As a consequence, this possibility is ruled out for knots with bridge number at most five and for positive braid closures.
We study complexities of 3-manifolds defined from triangulations, Heegaard splittings, and surgery presentations. We show that these complexities are related by linear inequalities, by presenting explicit geometric constructions. We also show that our linear inequalities are asymptotically optimal. Our results are used in [arXiv:1405.1805] to estimate Cheeger-Gromov $L^2$ $ρ$-invariants in terms of geometric group theoretic and knot theoretic data.
We derive a new exact sequence in the hat-version of Heegaard Floer homology. As a consequence we see a functorial connection between the invariant of Legendrian knots and the contact element. As an application we derive two vanishing results of the contact element making it possible to easily read off its vanishing out of a surgery presentation in suitable situations.
Let $M_n$ be a homology 3-sphere obtained by $\frac1n$-Dehn surgery along a $(p,q)$-torus knot. We consider a polynomial $σ_{(p,q,n)}(t)$ whose zeros are the inverses of the Reideimeister torsion of $M_n$ for $\mathit{SL}(2;\mathbb{C})$-irreducible representations. We give an explicit formula of this polynomial by using Tchebychev polynomials of the first kind. Further we also give a 3-term relations of these polynomials.
In this paper, we use Heegaard Floer homology to study reducible surgeries. In particular, suppose K is a non-cable knot in the three-sphere with an L-space surgery. If p-surgery on K is reducible, we show that p equals 2g(K)-1. This implies that any knot with an L-space surgery has at most one reducible surgery, a fact that we show additionally for any knot of genus at most two.
Consider a three dimensional cusped spherical $\mathrm{CR}$ manifold $M$ and suppose that the holonomy representation of $π_1(M)$ can be deformed in such a way that the peripheral holonomy is generated by a non-parabolic element. We prove that, in this case, there is a spherical $\mathrm{CR}$ structure on some Dehn surgeries of $M$. The result is very similar to R. Schwartz's spherical $\mathrm{CR}$ Dehn surgery theorem, but has weaker hypotheses and does not give the unifomizability of the structure. We apply our theorem in the case of the Deraux-Falbel structure on the Figure Eight knot comp…
This is an expository paper, in which we give a summary of some of the joint work of John Luecke and the author on Dehn surgery. We consider the situation where we have two Dehn fillings $M(α)$ and $M(β)$ on a given 3-manifold $M$, each containing a surface that is either essential or a Heegaard surface. We show how a combinatorial analysis of the graphs of intersection of the two corresponding punctured surfaces in $M$ enables one to find faces of these graphs which give useful topological information about $M(α)$ and $M(β)$, and hence, in certain cases, good upper bounds on the intersection …
Motivated by Clay and Watson's question on left-orderability of the fundamental group of the resultant space of an $r'$-surgery on the $(p, q)$-cable knots for $r' \in (pq-p-q,pq)$, this paper proves by elementary means that for specific pairs of $(p,q)$-cable knots of torus knots, $r' \in [pq-1,pq]$ gives a surgery yielding non-left orderable fundamental groups.
A combinatorial condition is obtained for when immersed or embedded incompressible surfaces in compact 3-manifolds with tori boundary components remain incompressible after Dehn surgery. A combinatorial characterisation of hierarchies is described. A new proof is given of the topological rigidity theorem of Hass and Scott for 3-manifolds containing immersed incompressible surfaces, as found in cubings of non-positive curvature.
When can one 3-manifold be transformed to another by a finite sequence of Dehn surgeries which are restricted to preserve the first homology of the manifolds ? What is the resulting equivalence relation on 3-manifolds ? What if the surgery circle is further restricted to lie more deeply in the lower central series of the fundamental group ? We answer these questions. It is shown that many of these questions have answers in terms of classical toplogical invariants. Relations with Heegard splittings and the Torelli group are discussed. This is also related to whether or not one 3-manifold may be…
In this paper we define a new invariant of the incomplete hyperbolic structures on a 1-cusped finite volume hyperbolic 3-manifold M, called the ortholength invariant. We show that away from a (possibly empty) subvariety of excluded values this invariant both locally parameterises equivalence classes of hyperbolic structures and is a complete invariant of the Dehn fillings of M which admit a hyperbolic structure. We also give an explicit formula for the ortholength invariant in terms of the traces of the holonomies of certain loops in M. Conjecturally this new invariant is intimately related to…
In a talk at the Cornell Topology Festival in 2005, W. Thurston discussed a graph which we call "The Big Dehn Surgery Graph", B. Here we explore this graph, particularly the link of S^3, and prove facts about the geometry and topology of B. We also investigate some interesting subgraphs and pose what we believe are important questions about B.
Using the combinatorial approach to Heegaard Floer homology we obtain a relatively easy formula for computation of hat Heegaard Floer homology for the three-manifold obtained by rational surgery on a knot K inside a homology sphere Y.
This largely technical paper is divided into two parts: part I: An account of P. Ozsvath and Z. Szabo's construction of the link surgery spectral sequence. There are no new results here, but this part slightly modifies and expands their proof and is included as an aid to part II. part II: Some modifications of the spectral sequence suitable for knot Floer homology; an exposition of the invariance of the spectral sequence, under a suitable equivalence, from alterations of the underlying bouquet; and an account of the morphisms induced on the spectral sequences by suitable four dimensional cobor…
Casson and Gordon gave the rectangle condition for strong irreducibility of Heegaard splittings [1]. We give a parity condition for irreducibility of Heegaard splittings of irreducible manifolds. As an application, we give examples of non-stabilized Heegaard splittings by doing a single Dehn twist.
We show that, for any given 3-manifold M, there are at most finitely many hyperbolic knots K in the 3-sphere and fractions p/q (with q > 22), such that M is obtained by p/q surgery along K. This is a corollary of the following result. If M is obtained by Dehn filling the cusps of a hyperbolic 3-manifold X, where each filling slope has length more than 2 π+ ε, then, for any given M and ε> 0, there are only finitely many possibilities for X and for the filling slopes. In this paper, we also investigate the length of boundary slopes, and sequences of negatively curved metrics on a given 3-manifol…
