Yoav Moriah

High distance knots

We construct knots in S-3 with Heegaard splittings of arbitrarily high distance, in any genus. As an application, for any positive integers t and b we find a tunnel number t knot in the three-sphere which has no (t, b) -decomposition.

Heegaard splittings of Seifert fibered spaces

SummaryIn this paper we give a classification theorem of genus two Heegaard splittings of Seifert fibered manifolds overS2 with three exceptional fibers, except for when two of the exceptional fibers hava the same invariants with opposite orientation.

On the free genus of knots

The class of knots consisting of twisted Whitehead doubles can have arbitrarily large free genus but all have genus 1.

Closed incompressible surfaces in complements of wide knots and links

Abstract We define the notion of wide knots (and links) and show that they contain closed incompressible nonboundary parallel surfaces in their complement. This is done by proving that these complements admit Heegaard splittings which are irreducible but weakly reducible, and using an extension of a result of Casson and Gordon. We then show that the class of wide knots and links is rather large, and that examples are easy to come by. We also show that the incompressible surfaces remain incompressible after most Dehn fillings.

On boundary primitive manifolds and a theorem of Casson–Gordon

Abstract In this paper it is shown that manifolds admitting minimal genus weakly reducible but irreducible Heegaard splittings contain an essential surface. This is an extension of a well-known theorem of Casson–Gordon to manifolds with non-empty boundary. The situation for non-minimal genus Heegaard splittings is also investigated and it is shown that boundary stabilizations are stabilizations for manifolds which are boundary primitive.

Heegaard splittings of knot exteriors

The goal of this paper is to offer a comprehensive exposition of the current knowledge about Heegaard splittings of exteriors of knots in the 3-sphere. The exposition is done with a historical perspective as to how ideas developed and by whom. Several new notions are introduced and some facts about them are proved. In particular the concept of a 1/n-primitive meridian. It is then proved that if a knot K in S^3 has a 1/n-primitive meridian; then nK = K#...#K, n-times has a Heegaard splitting of genus nt(K) + n which has a 1-primitive meridian. That is, nK is mu-primitive.

Tubed incompressible surfaces in knot and link complements

Abstract We prove that the complements of all knots and links in S 3 which have a 2n -plat projection with absolute value of all twist coefficients bigger than 2 contain closed embedded incompressible non-boundary parallel surfaces. These surfaces are obtained from essential planar meridional surfaces by tubing to one side along the knot or link. In the case of a knot it follows that these surfaces stay incompressible in all manifolds obtained by non-trivial surgery on the knot.

Manifolds with irreducible Heegaard splittings of high genus

Non-isotopic Heegaard splittings of non-minimal genus were known previously only for very special 3-manifolds. We show in this paper that they are in fact a widespread phenomenon in 3-manifold theory: We exhibit a large class of knots and manifolds obtained by Dehn surgery on these knots which admit such splittings. Many of the manifolds have irreducible Heegaard splittings of arbitrary large genus. All these splittings are horizontal and are isotopic, after one stabilization, to a multiple stabilization of certain canonical low genus vertical Heegaard splittings. ( 2000 Published by Elsevier Science Ltd. All rights reserved.

Closed incompressible surfaces in knot complements

In this paper we show that given a knot or link K in a 2n-plat projection with n ≥ 3 and m ≥ 5, where m is the length of the plat, if the twist coefficients ai,j all satisfy |ai,j | > 1 then S3 −N(K) has at least 2n− 4 nonisotopic essential meridional planar surfaces. In particular if K is a knot then S3−N(K) contains closed incompressible surfaces. In this case the closed surfaces remain incompressible after all surgeries except perhaps along a ray of surgery coefficients in Z⊕ Z.

Horizontal Dehn surgery and genericity in the curve complex

We introduce a general notion of ‘genericity’ for countable subsets of a space with Borel measure, and apply it to the set of vertices in the curve complex of a surface Σ, interpreted as subset of the space of projective measured laminations in Σ, equipped with its natural Lebesgue measure. We prove that, for any 3-manifold M , the set of curves c on a Heegaard surface Σ ⊂ M ,s uch that every non-trivial Dehn twist at c yields a Heegaard splitting of high distance, is generic in the set of all essential simple closed curves on Σ. Our definition of ‘genericity’ is different and more intrinsic than alternative such existing notions, given, for example, via random walks or via limits of quotients of finite sets.