## The size of spanning disks for polygonal knots (1999)

Citations: | 6 - 1 self |

### BibTeX

@TECHREPORT{Hass99thesize,

author = {Joel Hass and Jack Snoeyink and William P. Thurston},

title = {The size of spanning disks for polygonal knots},

institution = {},

year = {1999}

}

### OpenURL

### Abstract

For each integer n ≥ 1 we construct a closed unknotted Piecewise Linear curve Kn in R 3 having less than 11n edges with the property that any Piecewise Linear triangluated disk spanning the curve contains at least 2 n−1 triangles. 1 Introduction. We show the existence of a sequence of unknotted simple closed curves Kn in R 3 having the following properties: • The curve Kn is a polygon with at most 11n edges. • Any Piecewise Linear (PL) embedding of a triangulated disk into R 3 with

### Citations

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Three-Dimentional Geometry and Topology, Vol 1
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(Show Context)
Citation Context ... avoided by more topological arguments. The methods used here to construct the knots Kn and to establish their properties are based on ideas from the classification of diffeomorphisms of surfaces [2],=-=[9]-=-. ∗ Partially supported by NSF grant DMS-9704286. † Partially supported by grants from NSERC. 12 A braid, a diffeomorphism and a train track. To construct Kn, we begin with the 4-braid α = σ1σ −1 2 ,... |

79 |
Combinatorics of train-tracks
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(Show Context)
Citation Context ...re 1: The braid α. One tool for understanding diffeomorphisms of surfaces is invariant train tracks. We will use only elementary ideas from the theory of train tracks; for further discussion see [9], =-=[7]-=-. A train track is a 3-valent graph that is embedded on a surface with the edges (called tracks) embedded smoothly and the three tangent directions at a vertex (called a switch) lining up. Train track... |

40 | The number of Reidemeister moves needed for unknotting
- Hass, Lagarias
- 1998
(Show Context)
Citation Context ...by searching for embedded disks in the complement need to deal with disks containing exponentially many triangles. Thus the exponential bounds on the size of the normal disks that are used in [1],[3],=-=[4]-=-,[5],[6] cannot be replaced with polynomial bounds. Approaches to other problems, such as the word problem for 3-manifold groups, are generally based on a search for a disk with boundary a given curve... |

30 |
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(Show Context)
Citation Context ...n be avoided by more topological arguments. The methods used here to construct the knots Kn and to establish their properties are based on ideas from the classification of diffeomorphisms of surfaces =-=[2]-=-,[9]. ∗ Partially supported by NSF grant DMS-9704286. † Partially supported by grants from NSERC. 12 A braid, a diffeomorphism and a train track. To construct Kn, we begin with the 4-braid α = σ1σ −1... |

14 |
The computational complexity of knot and link problems, preprint
- Hass, Lagarias, et al.
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(Show Context)
Citation Context ...earching for embedded disks in the complement need to deal with disks containing exponentially many triangles. Thus the exponential bounds on the size of the normal disks that are used in [1],[3],[4],=-=[5]-=-,[6] cannot be replaced with polynomial bounds. Approaches to other problems, such as the word problem for 3-manifold groups, are generally based on a search for a disk with boundary a given curve, an... |

10 |
A trivial Knot whose Spanning Disks have exponential size
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(Show Context)
Citation Context ...ups, are generally based on a search for a disk with boundary a given curve, and our result may indicate why such searches seem to be computationally difficult. A result of this type was announced in =-=[8]-=-, but the geometric analysis suggested there seems to be difficult to establish rigorously. In particular, there is a mismatch between the base case and the inductive step that can be avoided by more ... |

7 |
Recognizing the unknot, preprint
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(Show Context)
Citation Context ...viality by searching for embedded disks in the complement need to deal with disks containing exponentially many triangles. Thus the exponential bounds on the size of the normal disks that are used in =-=[1]-=-,[3],[4],[5],[6] cannot be replaced with polynomial bounds. Approaches to other problems, such as the word problem for 3-manifold groups, are generally based on a search for a disk with boundary a giv... |

7 | Knots and braids: some algorithmic questions - Welsh - 1991 |

4 | The complexity of knots, in: Quo Vadis, Graph Theory - Welsh - 1993 |

3 |
On a problem in effective knot theory. Rendiconti dell Accademia dei Lincei 9
- Galatolo
- 1999
(Show Context)
Citation Context ...ity by searching for embedded disks in the complement need to deal with disks containing exponentially many triangles. Thus the exponential bounds on the size of the normal disks that are used in [1],=-=[3]-=-,[4],[5],[6] cannot be replaced with polynomial bounds. Approaches to other problems, such as the word problem for 3-manifold groups, are generally based on a search for a disk with boundary a given c... |