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.     

Weakly reducible Heegaard splittings of Seifert fibered spaces

We prove that for exceptional Seifert manifolds all weakly reducible Heegaard splittings are reducible. This provides the missing case for the Main Theorem in (Moriah and Schultens, to appear). It follows that for all orientable Seifert fibered spaces which fiber over an orientable base space, irreducible Heegaard splittings are either horizontal or vertical.     

Almost Alternating Diagrams and Fibered Links in S3

Let R be a Seifert surface obtained by applying Seifert's algorithmto a connected diagram D for a link L. In this paper, lettingD be almost alternating, we give a practical algorithm to determinewhether L is a fibered link and R is a fiber surface. We furthershow that L is a fibered link and R is a fiber surface for Lif and only if R is a Hopf plumbing, that is, a successive plumbingof a finite number of Hopf bands. It has been known for sometime that this is true if D is alternating, and we show thatit is not always true if D is 2-almost alternating. In the appendix,we partially answer C. Adams's open question concerning almostalternating diagrams. 2000 Mathematical Subject Classification:57M25.     

Weakly reducible Heegaard splittings of Seifert fibered spaces

We prove that for exceptional Seifert manifolds all weakly reducible Heegaard splittings are reducible. This provides the missing case for the Main Theorem in (Moriah and Schultens, to appear). It follows that for all orientable Seifert fibered spaces which fiber over an orientable base space, irreducible Heegaard splittings are either horizontal or vertical.     

Heegaard splittings of Seifert fibered spaces

Summary In this paper we give a classification theorem of genus two Heegaard splittings of Seifert fibered manifolds overS ² with three exceptional fibers, except for when two of the exceptional fibers hava the same invariants with opposite orientation.     

A Calculation of the Culler-shalen Seminorms Associated to Small Seifert Dehn Fillings

We determine the relationship between the multiplicities ofthe zeros of certain rational functions defined on the SL₂(C)and PSL₂C character varieties of knot exteriors. This leadsto a formula for the Culler–Shalen seminorms associatedto small Seifert Dehn fillings of such manifolds. 2000 MathematicsSubject Classification: 57M05, 57M27, 57R65.     

Invariant sets and measures of nonexpansive group automorphisms

We write a formula for the LMO invariant of a rational homology sphere presented as a rational surgery on a link inS ³. Our main tool is a careful use of the Århus integral and the (now proven) “Wheels” and “Wheeling” conjectures of B-N, Garoufalidis, Rozansky and Thurston. As steps, side benefits and asides we give explicit formulas for the values of the Kontsevich integral on the Hopf link and on Hopf chains, and for the LMO invariant of lens spaces and Seifert fibered spaces. We find that the LMO invariant does not separate lens spaces, is far from separating general Seifert fibered spaces, but does separate Seifert fibered spaces which are integral homology spheres.     

A theory of cobordism for non-spherical links

We define an equivalence relation, called algebraic cobordism, on the set of bilinear forms over the integers. When , we prove that two 2n - 1 dimensional, simple fibered links are cobordant if and only if they have algebraically cobordant Seifert forms. As an algebraic link is a simple fibered link, our criterion for cobordism allows us to study isolated singularities of complex hypersurfaces up to cobordism. Received: August 24, 1995     

The Neumann-Siebenmann invariant and Seifert surgery

We discuss some relations between the invariant originated in Fukumoto-Furuta and the Neumann-Siebenmann invariant for the Seifert rational homology 3-spheres. We give certain constraints on Seifert 3-manifolds to be obtained by surgery on knots in homology 3-spheres in terms of these invariants.Mathematics Subject Classification (2000): 57M27, 57N13, 57N10Dedicated to Professor Yukio Matsumoto for his 60^th birthday     

Cartesian product stabilization of 3-manifolds

The Cartesian product of a closed, orientable prime geometric 3-manifold and a closed orientable surface is unique except for the case of the Cartesian product of a special class of Seifert manifolds and a torus. The same type of uniqueness holds for stabilization of 3-manifolds by an n-dimensional torus. Cartesian squares of Seifert fibered 3-manifolds are completely classified.     

Heegaard genus and property {tau} for hyperbolic 3-manifolds

We show that any finitely generated non-elementary Kleiniangroup has a co-final family of finite index normal subgroupswith respect to which it has Property . As a consequence, anyclosed hyperbolic 3-manifold has a co-final family of finiteindex normal subgroups for which the infimal Heegaard gradientis positive. Received March 25, 2007.     

A theory of concordance for non-spherical 3-knots

Consider a closed connected oriented 3-manifold embedded in the -sphere, which is called a -knot in this paper. For two such knots, we say that their Seifert forms are spin concordant, if they are algebraically concordant with respect to a diffeomorphism between the 3-manifolds which preserves their spin structures. Then we show that two simple fibered 3-knots are geometrically concordant if and only if they have spin concordant Seifert forms, provided that they have torsion free first homology groups. Some related results are also obtained.

     

Quasi-Fuchsian Seifert surfaces

Let be a non fibered knot with hyperbolic complement. Given a Seifert surface of minimal genus for , we prove that it corresponds to a quasi-Fuchsian group, by showing it has no accidental parabolics. In particular, this shows that any lift of the Seifert surface to the universal cover is a quasi-disk and its limit set is a quasi-circle in the sphere at infinity. Received 8 November 1994; in final form 12 September 1995     

Taut Foliations in Punctured Surface Bundles, II

Given a fibered 3-manifold M, we investigate exactly which boundaryslopes can be realized by perturbing fibrations along productdiscs. Since these perturbed fibrations cap off to give tautfoliations in the corresponding surgery manifolds, we obtainsurgery information. For example, recall that a knot k is saidto satisfy Property P if no finite surgery along k yields asimply-connected 3-manifold. We show that if a non-trivial fiberedknot k S³ fails to satisfy Property P, then necessarily k ishyperbolic with degeneracy slope . When k is hyperbolic and (respectively, ), we show that the only candidate for a counterexample to Property P is surgery coefficient (respectively, . 2000 Mathematical Subject Classification: primary57M25; secondary 57R30.     

On the knot complement problem for non-hyperbolic knots

This paper explicitly provides two exhaustive and infinite families of pairs (M,k), where M is a lens space and k is a non-hyperbolic knot in M, which produces a manifold homeomorphic to M, by a non-trivial Dehn surgery. Then, we observe the uniqueness of such knot in such lens space, the uniqueness of the slope, and that there is no preserving homeomorphism between the initial and the final M's. We obtain further that Seifert fibered knots, except for the axes, and satellite knots are determined by their complements in lens spaces. An easy application shows that non-hyperbolic knots are determined by their complement in atoroidal and irreducible Seifert fibered 3-manifolds.     

Comparing Heegaard and JSJ structures of orientable 3-manifolds

The Heegaard genus of an irreducible closed orientable -manifold puts a limit on the number and complexity of the pieces that arise in the Jaco-Shalen-Johannson decomposition of the manifold by its canonical tori. For example, if of the complementary components are not Seifert fibered, then . This generalizes work of Kobayashi. The Heegaard genus also puts explicit bounds on the complexity of the Seifert pieces. For example, if the union of the Seifert pieces has base space and exceptional fibers, then .

     

Reidemeister torsion for linear representations and Seifert surgery on knots

We study an invariant of a 3-manifold which consists of Reidemeister torsion for linear representations which pass through a finite group. We show a Dehn surgery formula on this invariant and compute that of a Seifert manifold over S². As a consequence we obtain a necessary condition for a result of Dehn surgery along a knot to be Seifert fibered, which can be applied even in a case where abelian Reidemeister torsion gives no information.     

The combinatorics of some tetrahedron manifolds

We study a family of closed connected orientable 3-manifolds (which are examples of tetrahedron manifolds) obtained by pairwise identifications of the boundary faces of a standard tetrahedron. These manifolds generalize those considered in previous papers due to Grasselli, Piccarreta, Molnár and Sieradski. Then we completely describe our tetrahedron manifolds in terms of Seifert fibered spaces, and determine their Seifert invariants. Moreover, we obtain different representations of our manifolds as 2-fold coverings, and give examples of non-equivalent knots with the same tetrahedron manifold as 2-fold branched covering space.     

Finite covers of random 3-manifolds

A 3-manifold is Haken if it contains a topologically essential surface. The Virtual Haken Conjecture posits that every irreducible 3-manifold with infinite fundamental group has a finite cover which is Haken. In this paper, we study random 3-manifolds and their finite covers in an attempt to shed light on this difficult question. In particular, we consider random Heegaard splittings by gluing two handlebodies by the result of a random walk in the mapping class group of a surface. For this model of random 3-manifold, we are able to compute the probabilities that the resulting manifolds have finite covers of particular kinds. Our results contrast with the analogous probabilities for groups coming from random balanced presentations, giving quantitative theorems to the effect that 3-manifold groups have many more finite quotients than random groups. The next natural question is whether these covers have positive betti number. For abelian covers of a fixed type over 3-manifolds of Heegaard genus 2, we show that the probability of positive betti number is 0.In fact, many of these questions boil down to questions about the mapping class group. We are led to consider the action of the mapping class group of a surface Σ on the set of quotients π₁(Σ)→Q. If Q is a simple group, we show that if the genus of Σ is large, then this action is very mixing. In particular, the action factors through the alternating group of each orbit. This is analogous to Goldman’s theorem that the action of the mapping class group on the SU(2) character variety is ergodic. Mathematics Subject Classification (2000) 57M50, 57N10     

Chord Diagrams and Coxeter Links

The paper presents a construction of fibered links (K, ) outof chord diagrams L. Let be the incidence graph of L. Undercertain conditions on L the symmetrized Seifert matrix of (K,) equals the bilinear form of the simply-laced Coxeter system(W, S) associated to and the monodromy of (K, ) equals minusthe Coxeter element of (W, S). Lehmer's problem is solved forthe monodromy of these Coxeter links.