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- Martin Scharlemann, Abigail Thompson
- 2008

It has been conjectured that the geometric invariant of knots in 3–space called the width is nearly additive. That is, letting w(K) ∈ 2N denote the width of a knot K ⊂ 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… (More)

- Geometry Topology, G G G G G G G G G G G G G G G, Hiroshi Goda, Martin Scharlemann, Abigail Thompson, Cameron Gordon +2 others
- 2000

T T T T T T T T T T T T T T T Abstract 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… (More)

It has been postulated that autism spectrum disorder is underpinned by an 'atypical connectivity' involving higher-order association brain regions. To test this hypothesis in a large cohort of adults with autism spectrum disorder we compared the white matter networks of 61 adult males with autism spectrum disorder and 61 neurotypical controls, using two… (More)

- Joel Hass, Abigail Thompson, William Thurston
- 2008

For each g ≥ 2 there is a 3-manifold with two genus g Heegaard splittings that require g stabilizations to become equivalent. Previously known examples required at most one stabilization. Control of families of Heegaard surfaces is obtained through a deformation to harmonic maps.

- Martin Scharlemann, Abigail Thompson
- 2002

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 ∈ Q/2Z, 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,… (More)

- Martin Scharlemann, Abigail Thompson
- 2001

We study trivalent graphs in S 3 whose closed complement is a genus two handlebody. We show that such a graph, when put in thin position, has a level edge connecting two vertices.

- Jesse Johnson, Abigail Thompson
- 2006

We show that for any natural number n, there is a tunnel number one knot in S 3 which is not (1, n).

- Martin Scharlemann, Abigail Thompson
- 2008

It is shown, using sutured manifold theory, that if there are any 2-component counterexamples to the Generalized Property R Conjecture, then any knot of least genus among components of such counterexamples is not a fibered knot. The general question of what fibered knots might appear as a component of such a counterexample is further considered; much can be… (More)

- Martin Scharlemann, Abigail Thompson
- 2008

Suppose F is a compact orientable surface, K is a knot in F × I, and (F × I)surg is the 3-manifold obtained by some non-trivial surgery on K. If F × {0} compresses in (F × I)surg, then there is an annulus in F × I with one end K and the other end an essential simple closed curve in F × {0}. Moreover, the end of the annulus at K determines the surgery slope.… (More)

- Jesse Johnson, Abigail Thompson
- 2007

We show that the bridge number of a t bridge knot in S 3 with respect to an unknotted genus t surface is bounded below by a function of the distance of the Heegaard splitting induced by the t bridges. It follows that for any natural number n, there is a tunnel number one knot in S 3 that is not (1, n).