# Distance and bridge position

@article{Bachman2003DistanceAB, title={Distance and bridge position}, author={D. Bachman and S. Schleimer}, journal={Pacific Journal of Mathematics}, year={2003}, volume={219}, pages={221-235} }

J. Hempel's denition of the distance of a Heegaard surface generalizes to a notion of complexity for any knot that is in bridge position with respect to a Heegaard surface. Our main result is that the distance of a knot in bridge position is bounded above by twice the genus, plus the number of boundary components, of an essential surface in the knot complement. As a consequence knots constructed via suciently high powers of pseudo-Anosov maps have minimal bridge presentations which are thin.

#### 56 Citations

Distortion and the bridge distance of knots

- Mathematics
- 2017

We extend techniques due to Pardon to show that there is a lower bound on the distortion of a knot in $\mathbb{R}^3$ proportional to the minimum of the bridge distance and the bridge number of the… Expand

Bridge and pants complexities of knots

- Mathematics, Computer Science
- J. Lond. Math. Soc.
- 2013

We modify an approach of Johnson to define the distance of a bridge splitting of a knot in a 3-manifold using the dual curve complex and pants complex of the bridge surface. This distance can be used… Expand

Distance of Heegaard splittings of knot complements

- Mathematics
- 2007

Let K be a knot in a closed orientable irreducible 3-manifold M and let P be a Heegaard splitting of the knot complement of genus at least two. Suppose Q is a bridge surface for K. Then either… Expand

ARC DISTANCE EQUALS LEVEL NUMBER

- Mathematics
- 2009

Let K be a knot in 1-bridge position with respect to a genus-g Heegaard surface that splits a 3-manifold M into two handlebodies V and W. One can move K by isotopy keeping K∩V in V and K∩W in W so… Expand

AN ESTIMATE OF HEMPEL DISTANCE FOR BRIDGE SPHERES

- Mathematics
- 2015

Tomova (8) gave an upper bound for the distance of a bridge surface for a knot with two different bridge positions in a 3-manifold. In this paper, we show that the result of Tomova (8, Theorem 10.3)… Expand

Thin position for knots in a 3-manifold

- Mathematics
- 2009

We extend the notion of a thin position of a link in a 3-manifold with respect to a generalized
Heegaard splitting introduced in Hayashi and Shimokawa (Pacific J. Math. 197 (2001) 301�324),
to take… Expand

Bridge distance, Heegaard genus, and Exceptional Surgeries

- Mathematics
- 2012

We demonstrate a lower bound on the genus of an essential surface or Heegaard surface in a 3-manifold obtained by non-trivial surgery on a link in terms of the bridge distance of a bridge surface for… Expand

COMPLEXITY OF OPEN BOOK DECOMPOSITIONS VIA ARC COMPLEX

- Mathematics
- 2010

Based on Hempel's distance of a Heegaard splitting, we define a certain kind of complexity of an open book decomposition, called a translation distance, by using the arc complex of its fiber surface.… Expand

Exceptional and cosmetic surgeries on knots

- Mathematics
- 2012

We show that the bridge distance of a knot determines a lower bound on the genera of essential surfaces and Heegaard surfaces in the manifolds that result from non-trivial Dehn surgeries on the knot.… Expand

RECTANGLE CONDITION AND A FAMILY OF ALTERNATING 3-BRIDGE KNOTS

- Mathematics
- 2014

In this paper, we define the rectangle condition for n-bridge presentation of knots whose definition is analogous to the definition of the rectangle condition for Heegaard splittings of 3-manifolds.… Expand

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