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Martin Scharlemann

Publications and source records attributed to Martin Scharlemann.

At least 37 records · Page 2Linked to original sources
arXiv2008

Surgery on a knot in (Surface x I)

Suppose F is a compact orientable surface, K is a knot in F x I, and N is the 3-manifold obtained by some non-trivial surgery on K. If F x {0} compresses in N, then there is an annulus in F x I with one end K and the other end an essential simple closed curve in F x {0}. Moreover, the end of the annulus at K determines the surgery slope. An application: suppose M is a compact orientable 3-manifold that fibers over the circle. If surgery on a knot K in M yields a reducible manifold, then either: the projection of K to S^1 has non-trivial winding number; or K lies in a ball; or K lies in a fiber; or K is a cabled knot.

math.GT↗
arXiv2006

Conway products and links with multiple bridge surfaces

Suppose a link K in a 3-manifold M is in bridge position with respect to two different bridge surfaces P and Q, both of which are c-weakly incompressible in the complement of K. Then either P and Q can be properly isotoped to intersect in a nonempty collection of curves that are essential (including non-meridional) on both surfaces, or K is a Conway product with respect to an incompressible Conway sphere that naturally decomposes both P and Q into bridge surfaces for the respective factor link(s).

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arXiv2006

Generalized Property R and the Schoenflies Conjecture

There is a relation between the generalized Property R Conjecture and the Schoenflies Conjecture that suggests a new line of attack on the latter. The approach gives a quick proof of the genus 2 Schoenflies Conjecture and suffices to prove the genus 3 case, even in the absence of new progress on the generalized Property R Conjecture.

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arXiv2005

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.

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arXiv2005

Proximity in the curve complex: boundary reduction and bicompressible surfaces

Suppose N is a compressible boundary component of a compact orientable irreducible 3-manifold M and Q is an orientable properly embedded essential surface in M in which each component is incident to N and no component is a disk. Let VN and QN denote respectively the sets of vertices in the curve complex for N represented by boundaries of compressing disks and by boundary components of Q. Theorem: Suppose Q is essential in M, then the distance d(VN, QN) \leq 1 - χ(Q). Hartshorn showed that an incompressible surface in a closed 3-manifold puts a limit on the distance of any Heegaard splitting. The theorem above provides a version of Hartshorn's result for merely compact 3-manifolds. In a similar spirit, here is the main result: Theorem: Suppose Q is connected, separating and compresses on both its sides, but not by way of disjoint disks, then either d(VN, QN) \leq 1 - χ(Q) or Q is obtained from two nested connected boundary-parallel surfaces by a vertical tubing. Forthcoming work with M. Tomova (math.GT/0501140) will show how an augmented version of this theorem leads to the same conclusion as in Hartshorn's theorem, not from an essential surface but from an alternate Heegaard surface. That is, if Q is a Heegaard splitting of a compact M then then no other irreducible Heegaard splitting of M has distance greater than twice the genus of Q.

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arXiv2004

On the additivity of knot width

It has been conjectured that the geometric invariant of knots in 3-space called the width is nearly additive. That is, letting w(K) in N denote the width of a knot K in S^3, the conjecture is that w(K # K') = w(K) + w(K') - 2. We give an example of a knot K_1 so that for K_2 any 2-bridge knot, it appears that w(K_1 # K_2) = w(K_1), contradicting the conjecture.

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arXiv2003

Surfaces, submanifolds, and aligned Fox reimbedding in non-Haken 3-manifolds

Understanding non-Haken 3-manifolds is central to many current endeavors in 3-manifold topology. We describe some results for closed orientable surfaces in non-Haken manifolds, and extend Fox's theorem for submanifolds of the 3-sphere to submanifolds of general non-Haken manifolds. In the case where the submanifold has connected boundary, we show also that the boundary-connected sum decomposition of the submanifold can be aligned with such a structure on the submanifold's complement.

math.GT↗
arXiv2003

Automorphisms of the 3-sphere that preserve a genus two Heegaard splitting

An updated proof of a 1933 theorem of Goeritz, exhibiting a finite set of generators for the group of automorphisms of the 3-sphere that preserve a genus two Heegaard splitting. The group is analyzed via its action on a certain connected 2-complex. (The analogous problem for higher genus Heegaard splittings appears to remain unresolved.)

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arXiv2003

3-manifolds with planar presentations and the width of satellite knots

We consider compact 3-manifolds M having a submersion h to R in which each generic point inverse is a planar surface. The standard height function on a submanifold of the 3-sphere is a motivating example. To (M, h) we associate a connectivity graph G. For M in the 3-sphere, G is a tree if and only if there is a Fox reimbedding of M which carries horizontal circles to a complete collection of complementary meridian circles. On the other hand, if the connectivity graph of the complement of M is a tree, then there is a level-preserving reimbedding of M so that its complement is a connected sum of handlebodies. Corollary: The width of a satellite knot is no less than the width of its pattern knot. In particular, the width of K_1 # K_2 is no less than the maximum of the widths of K_1 and K_2.

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arXiv2001

The Goda-Teragaito conjecture: an overview

In math.GT/0106017 it was shown that thin position on Heegaard spines can be a useful tool for analyzing the topology of knots in 3-space. The proof there (specifically, of the Goda-Teragaito conjecture) requires masses of technical detail; it is easy to lose track of the underlying ideas. The present paper gives an overview of the argument, stripped of technical detail and emphasizing the mathematical background that seems naturally to motivate the proof.

math.GT↗
arXiv2001

There are no unexpected tunnel number one knots of genus one

The only knots that are tunnel number one and genus one are those that are already known: 2-bridge knots obtained by plumbing together two unknotted annuli and the satellite examples classified by Eudave-Munoz and by Morimoto-Sakuma. This confirms a conjecture first made by Goda and Teragaito.

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arXiv2000

Unknotting tunnels and Seifert surfaces

Let $K$ be a knot with an unknotting tunnel $γ$ and suppose that $K$ is not a 2-bridge knot. There is an invariant $ρ= p/q \in \mathbb{Q}/2 \mathbb{Z}$, $p$ odd, defined for the pair $(K, γ)$. The invariant $ρ$ has interesting geometric properties: It is often straightforward to calculate; e. g. for $K$ a torus knot and $γ$ an annulus-spanning arc, $ρ(K, γ) = 1$. Although $ρ$ is defined abstractly, it is naturally revealed when $K \cup γ$ is put in thin position. If $ρ\neq 1$ then there is a minimal genus Seifert surface $F$ for $K$ such that the tunnel $γ$ can be slid and isotoped to lie on $F$. One consequence: if $ρ(K, γ) \neq 1$ then $genus(K) > 1$. This confirms a conjecture of Goda and Teragaito for pairs $(K, γ)$ with $ρ(K, γ) \neq 1$.

math.GT↗
arXiv2000

Levelling an unknotting tunnel

It is a consequence of theorems of Gordon-Reid [Tangle decompositions of tunnel number one knots and links, J. Knot Theory and its Ramifications, 4 (1995) 389-409] and Thompson [Thin position and bridge number for knots in the 3-sphere, Topology, 36 (1997) 505-507] 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 [A note on unknotting tunnels for 2-bridge knots, Bulletin of Faculty of Engineering Takushoku University, 3 (1992) 219-225], who showed that the (now known) classification of unknotting tunnels for 2-bridge knots would follow quickly if it were known that any unknotting tunnel can be made level.

math.GT↗
arXiv1999

Genus two Heegaard splittings of orientable three-manifolds

It was shown by Bonahon-Otal and Hodgson-Rubinstein that any two genus-one Heegaard splittings of the same 3-manifold (typically a lens space) are isotopic. On the other hand, it was shown by Boileau, Collins and Zieschang that certain Seifert manifolds have distinct genus-two Heegaard splittings. In an earlier paper, we presented a technique for comparing Heegaard splittings of the same manifold and, using this technique, derived the uniqueness theorem for lens space splittings as a simple corollary. Here we use a similar technique to examine, in general, ways in which two non-isotopic genus-two Heegard splittings of the same 3-manifold compare, with a particular focus on how the corresponding hyperelliptic involutions are related.

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