Abigail Thompson

The disjoint curve property and genus 2 manifolds

Abstract Genus 2 manifolds are a convenient and accessible place to introduce an interesting condition on Heegaard splittings, called the disjoint curve property . This paper will describe the disjoint curve property and its ramifications for understanding genus 2 manifolds, and use it to find a necessary condition for a genus 2 manifold to be hyperbolic.

Stabilization of Heegaard splittings

A genus‐g Heegaard splitting of a 3‐manifold M is a decomposition of M into two genus‐g handlebodies with a common boundary. It is described by an ordered triple .H1;H2;S/ where each of H1;H2 is a handlebody and the two handlebodies intersect along their common boundary S , called a Heegaard surface. An orientation on S is determined by @H1 , and an equivalent definition of a Heegaard splitting is given by an oriented surface S in M whose complement consists of two handlebodies. Two Heegaard splittings .H1;H2;S/ and .H 0 1 ;H 0 2 ;S 0 / of M are said to be equivalent

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.

Unknotting Tunnels and Seifert Surfaces

Let

ON TUNNEL NUMBER ONE KNOTS THAT ARE NOT (1, n)

K

Thinning Genus Two Heegaard Spines in S3

be a knot with an unknotting tunnel

Algorithmic recognition of 3-manifolds

\gamma

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

and suppose that

Surgery on a knot in (surface × I)

K

A polynomial invariant of graphs in 3-manifolds

is not a