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. 

Figures from this paper

Distortion and the bridge distance of knots
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 theExpand
Bridge and pants complexities of knots
  • A. Zupan
  • 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 usedExpand
Distance of Heegaard splittings of knot complements
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 eitherExpand
ARC DISTANCE EQUALS LEVEL NUMBER
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 soExpand
AN ESTIMATE OF HEMPEL DISTANCE FOR BRIDGE SPHERES
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
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 takeExpand
Bridge distance, Heegaard genus, and Exceptional Surgeries
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 forExpand
COMPLEXITY OF OPEN BOOK DECOMPOSITIONS VIA ARC COMPLEX
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
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
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
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 20 REFERENCES
3-Manifolds as viewed from the curve complex ☆
Abstract A Heegaard diagram for a 3-manifold is regarded as a pair of simplexes in the complex of curves on a surface and a Heegaard splitting as a pair of subcomplexes generated by the equivalentExpand
Geometry of the complex of curves I: Hyperbolicity
The Complex of Curves on a Surface is a simplicial complex whose vertices are homotopy classes of simple closed curves, and whose simplices are sets of homotopy classes which can be realizedExpand
Geometry of the complex of curves II: Hierarchical structure
Abstract. ((Without Abstract)).
Heegaard splittings: the distance complex and the stabilization conjecture
  • Heegaard splittings: the distance complex and the stabilization conjecture
  • 1999
Minsky,“Geometry of the complex of curves, I: Hyperbolicity”, Invent
  • 103–149.MR
  • 1999
On unknotting tunnels for knots
...
1
2
...