Martin Scharlemann

Producing reducible 3-manifolds by surgery on a knot

IT HAS long been conjectured that surgery on a knot in S3 yields a reducible 3-manifold if and only if the knot is cabled, with the cabling annulus part of the reducing sphere (cf. [7.8, 9, 10, 111). One may regard the Poenaru conjecture (solved in [S]) as a special case of the above. More generally, one can ask when surgery on a knot in an arbitary 3-manifold A4 produces a reducible 3-manifold M’. But this problem is too complex, since, dually, it asks which knots in which manifolds arise from surgery on reducible 3-manifolds. In this paper we are able to show, approximately, that if M itself either contains a summand not a rational homology sphere or is a-reducible, and M’ is reducible, then k must have been cabled and the surgery is via the slope of the cabling annulus. Thus the result stops short of proving the conjecture for M = S3, but (see below) does suffice to prove the conjecture for satellite knots. The results here are broader than this; for a context recall the main result of [3]: GABAI’S THEOREM. Let k be a knot in M = D2 x S’ with nonzero wrapping number. If_ci’ is a manifold obtained by non-trivial surgery on k then one of the following must hold: (1) M = D2 x S’ = M’ and both k and k’ are 0 or l-bridge braids. (2) M’ = WI # W,, where W2 is a closed 3-manifold and H, ( W,) isjnite and non-tritial. (3) M’ is irreducible and aM ‘ is incompressible. It is this theorem we generalize to many other manifolds M. We also examine case (2) and show that it only arises (i.e. M’ is only reducible) if k is cabled with cabling annulus having the slope of the surgery. (For a detailed look at case (I), see [6] or Cl].) Specifically we have:

Alternate Heegaard genus bounds distance

Suppose M is a compact orientable irreducible 3-manifold with Heegaard splitting surfaces P and Q. Then either Q is isotopic to a possibly stabilized or boundary-stabilized copy of P or the distance d(P)<= 2genus (Q). More generally, if P and Q are bicompressible but weakly incompressible connected closed separating surfaces in M then either P and Q can be well-separated or P and Q are isotopic or d(P)<= 2genus (Q).

Comparing Heegaard splittings of non-haken 3-manifolds

Abstract Cerf theory can be used to compare two strongly irreducible Heegaard splittings of the same closed orientable 3-manifold. Any two splitting surfaces can be isotoped so that they intersect in a non-empty collection of curves, each of which is essential in both splitting surfaces. More generally, there are interesting isotopies of the splitting surfaces during which this intersection property is preserved. As sample applications we give new proofs of Waldhausens theorem that Heegaard splittings of S3 are standard, and of Bonahon and Otals theorem that Heegaard splittings of lens spaces are standard. We also present a solution to the stabilization problem for irreducible non-Haken 3-manifolds: If p ⩽ q are the genera of two splittings of such a manifold, then there is a common stabilization of genus 5p + 8q − 9.

Chapter 18 – Heegaard Splittings of Compact 3-Manifolds

1. Background 2 2. Heegaard splittings and their guises 3 2.1. Splittings from triangulations 3 2.2. Splitting 3-manifolds with boundary 3 2.3. Splittings as handle decompositions Heegaard diagrams 4 2.4. Splittings as Morse functions and as sweep-outs 6 3. Structures on Heegaard splittings 8 3.1. Stabilization 8 3.2. Reducible splittings 9 3.3. Weakly reducible splittings 12 4. Heegaard splittings in nature: Seifert manifolds 15 4.1. Vertical splittings 17 4.2. Horizontal splittings 17 5. Connections with group presentations 19 6. Uniqueness 21 6.1. The 3-sphere 21 6.2. Seifert manifolds 22 6.3. Genus and the Casson-Gordon examples 23 6.4. Other uniqueness results 24 7. The stabilization problem 24 8. Normal surfaces and decision problems 31 8.1. Normal surfaces and Heegaard splittings 31 8.2. Special case: Normal surfaces in a triangulation 33 References 36

Local detection of strongly irreducible Heegaard splittings

Abstract Let S be a Heegaard splitting surface of a compact orientable 3-manifold M . If S is strongly irreducible, the manner in which it can intersect a ball or a solid torus in M is very constrained and the allowable configurations are simple and useful. Splitting surfaces not conforming to these simple local pictures must be weakly reducible.

Tunnel number one knots satisfy the poenaru conjecture

Abstract It is shown that tunnel number one knots satisfy the Poenaru conjecture and so have Property R . As a sidelight they are also shown to be doubly prime.

THE TUNNEL NUMBER OF THE SUM OF n KNOTS IS AT LEAST n

We prove that the tunnel number of the sum of n knots is at least n.

Uniqueness of bridge surfaces for 2-bridge knots

Any 2-bridge knot in S3 has a bridge sphere from which any other bridge surface can be obtained by stabilization, meridional stabilization, perturbation and proper isotopy.

Levelling an unknotting tunnel

It is a consequence of theorems of Gordon{Reid [4] and Thompson [8] that a tunnel number one knot, if put in thin position, will also be in bridge position. We show that in such a thin presentation, the tunnel can be made level so that it lies in a level sphere. This settles a question raised by Morimoto [6], who showed that the (now known) classication of unknotting tunnels for 2{bridge knots would follow quickly if it were known that any unknotting tunnel can be made level.

Lecture Notes on Generalized Heegaard Splittings

These notes grew out of a lecture series given at RIMS in the summer of 2001. The lecture series was aimed at a broad audience that included many graduate students. Its purpose lay in familiarizing the audience with the basics of 3-manifold theory and introducing some topics of current research. The first portion of the lecture series was devoted to standard topics in the theory of 3-manifolds. The middle portion was devoted to a brief study of Heegaaard splittings and generalized Heegaard splittings. The latter portion touched on a brand new topic: fork complexes.