# Links, bridge number, and width trees

@article{He2020LinksBN, title={Links, bridge number, and width trees}, author={Qidong He and Scott A. Taylor}, journal={arXiv: Geometric Topology}, year={2020} }

To each link $L$ in $S^3$ we associate a collection of certain labelled directed trees, called width trees. We interpret some classical and new topological link invariants in terms of these width trees and show how the geometric structure of the width trees can bound the values of these invariants from below. We also show that each width tree is associated with a knot in $S^3$ and that if it also meets a high enough "distance threshold" it is, up to a certain equivalence, the unique width tree… Expand

#### References

SHOWING 1-10 OF 57 REFERENCES

Natural properties of the trunk of a knot

- Mathematics
- 2016

The trunk of a knot in $S^3$, defined by Makoto Ozawa, is a measure of geometric complexity similar to the bridge number or width of a knot. We prove that for any two knots $K_1$ and $K_2$, we have… Expand

On the tree-width of knot diagrams

- Mathematics, Computer Science
- J. Comput. Geom.
- 2019

We show that a small tree-decomposition of a knot diagram induces a small sphere-decomposition of the corresponding knot. This, in turn, implies that the knot admits a small essential planar… Expand

High Distance Bridge Surfaces

- Mathematics
- 2012

Given integers b, c, g, and n, we construct a manifold M containing a c-component link L so that there is a bridge surface Sigma for (M,L) of genus g that intersects L in 2b points and has distance… Expand

Additive invariants for knots, links and graphs in 3–manifolds

- Mathematics
- Geometry & Topology
- 2018

We define two new families of invariants for (3-manifold, graph) pairs which detect the unknot and are additive under connected sum of pairs and (-1/2)-additive under trivalent vertex sum of pairs.… Expand

Height, trunk and representativity of knots

- Mathematics
- Journal of the Mathematical Society of Japan
- 2019

In this paper, we investigate three geometrical invariants of knots, the height, the trunk and the representativity.
First, we give a conterexample for the conjecture which states that the height is… Expand

Thin position for knots, links, and graphs in 3-manifolds

- Mathematics
- 2016

We define a new notion of thin position for a graph in a 3-manifold which combines the ideas of thin position for manifolds first originated by Scharlemann and Thompson with the idea of thin position… Expand

ON THE TREEWIDTH OF KNOT DIAGRAMS

- 2019

We show that a small tree-decomposition of a knot diagram induces a small sphere-decomposition of the corresponding knot. This, in turn, implies that the knot admits a small essential planar… Expand

Bridge distance and plat projections

- Mathematics
- 2013

We calculate the bridge distance for $m$-bridge knots/links in the $3$-sphere with sufficiently complicated $2m$-plat projections. In particular we show that if the underlying braid of the plat has… Expand

On the additivity of knot width

- Mathematics
- 2004

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… Expand

Flipping bridge surfaces and bounds on the stable bridge number

- Mathematics
- 2011

We show that if K is a knot in S 3 and U is a bridge sphere for K with high distance and 2n punctures, the number of perturbations of K required to interchange the two balls bounded by U via an… Expand