## Unknot diagrams requiring a quadratic number of Reidemeister moves to untangle (2007)

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Citations: | 4 - 1 self |

### BibTeX

@MISC{Hass07unknotdiagrams,

author = {Joel Hass and Tahl Nowik},

title = {Unknot diagrams requiring a quadratic number of Reidemeister moves to untangle},

year = {2007}

}

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### Abstract

We present a sequence of diagrams of the unknot for which the minimum number of Reidemeister moves required to pass to the trivial diagram is quadratic with respect to the number of crossings. These bounds apply both in S 2 and in R 2. 1

### Citations

67 | The knot book. An elementary introduction to the mathematical theory of knots - Adams - 1994 |

35 | The number of Reidemeister moves needed for unknotting
- Hass, Lagarias
(Show Context)
Citation Context ...n in Figure 1. The number of such moves required to connect two equivalent diagrams is difficult to estimate. An exponential upper bound is obtained ∗ Supported in part by NSF grant DMS 3289292. 1sin =-=[6]-=-, where it is shown that there is a positive constant c such that given an unknot diagram D with n crossings, no more than 2 cn Reidemeister moves are required to transform D to the trivial knot diagr... |

27 |
On types of knotted curves
- Alexander, Briggs
(Show Context)
Citation Context ... recovered, up to isotopy, by constructing a curve with the overcrossing arcs pushed slightly above the plane of the diagram and the remainder of the diagram lying in this plane. Alexander and Briggs =-=[2]-=- and independently Reidemeister [11] showed that two diagrams of the same knot can be connected through a sequence of moves of three types, commonly called Reidemeister moves, shown in Figure 1. The n... |

12 | Invariants of knot diagrams and relations among Reidemeister moves
- Östlund
(Show Context)
Citation Context ...uire somehow bounding from below the running time of all possible procedures. It is quite difficult to get lower bounds even for a particular pair of equivalent diagrams, as seen in [4], [5], [9] and =-=[10]-=-. We note that examples are constructed in [8] that show that it may require exponentially many faces to construct a PL spanning disk for an unknotted polygon. However these examples can be transforme... |

9 | W.P.: The size of spanning disks for polygonal curves
- Hass, Snoeyink, et al.
- 2003
(Show Context)
Citation Context ...me of all possible procedures. It is quite difficult to get lower bounds even for a particular pair of equivalent diagrams, as seen in [4], [5], [9] and [10]. We note that examples are constructed in =-=[8]-=- that show that it may require exponentially many faces to construct a PL spanning disk for an unknotted polygon. However these examples can be transformed to the trivial diagram using only a linear n... |

6 | S.: A lower bound for the number of Reidemeister moves of type III
- Carter, Elhamdadi, et al.
- 2006
(Show Context)
Citation Context ..., lower bounds require somehow bounding from below the running time of all possible procedures. It is quite difficult to get lower bounds even for a particular pair of equivalent diagrams, as seen in =-=[4]-=-, [5], [9] and [10]. We note that examples are constructed in [8] that show that it may require exponentially many faces to construct a PL spanning disk for an unknotted polygon. However these example... |

5 | Every Reidemeister move is needed for each knot type
- Hagge
- 2006
(Show Context)
Citation Context ...er bounds require somehow bounding from below the running time of all possible procedures. It is quite difficult to get lower bounds even for a particular pair of equivalent diagrams, as seen in [4], =-=[5]-=-, [9] and [10]. We note that examples are constructed in [8] that show that it may require exponentially many faces to construct a PL spanning disk for an unknotted polygon. However these examples can... |

5 | T.: Invariants of knot diagrams
- Hass, Nowik
- 2008
(Show Context)
Citation Context ...emeister moves required to pass from D to E. One may consider this notion in either S 2 or R 2 , and our result will hold in both settings. Our main tool is an invariant of knot diagrams developed in =-=[7]-=-, and used there to obtain new linear lower bounds on the Reidemeister distance. 22 The diagrams Let U denote the trivial knot diagram. We will present a sequence Dn of diagrams of the unknot, for wh... |

4 |
A lower bound for the number of Reidemeister moves for unknotting
- Hayashi
- 2006
(Show Context)
Citation Context ...unds require somehow bounding from below the running time of all possible procedures. It is quite difficult to get lower bounds even for a particular pair of equivalent diagrams, as seen in [4], [5], =-=[9]-=- and [10]. We note that examples are constructed in [8] that show that it may require exponentially many faces to construct a PL spanning disk for an unknotted polygon. However these examples can be t... |

1 |
Nowik: “Invariants of knot diagrams” arXiv:0708.2509 5
- Hass, T
(Show Context)
Citation Context ...emeister moves required to pass from D to E. One may consider this notion in either S 2 or R 2 , and our result will hold in both settings. Our main tool is an invariant of knot diagrams developed in =-=[7]-=-, and used there to obtain new linear lower bounds on the Reidemeister distance. 2 The diagrams Let U denote the trivial knot diagram. We will present a sequence Dn of diagrams of the unknot, for whic... |

1 |
The Size of Spanning Disks for
- Hass, Snoeyink, et al.
(Show Context)
Citation Context ...me of all possible procedures. It is quite difficult to get lower bounds even for a particular pair of equivalent diagrams, as seen in [4], [5], [9] and [10]. We note that examples are constructed in =-=[8]-=- that show that it may require exponentially many faces to construct a PL spanning disk for an unknotted polygon. However these examples can be transformed to the trivial diagram using only a linear n... |

1 |
7–23. email: hass@math.ucdavis.edu tahl@math.biu.ac.il address
- Sem, Hamburg
- 1926
(Show Context)
Citation Context ...ructing a curve with the overcrossing arcs pushed slightly above the plane of the diagram and the remainder of the diagram lying in this plane. Alexander and Briggs [2] and independently Reidemeister =-=[11]-=- showed that two diagrams of the same knot can be connected through a sequence of moves of three types, commonly called Reidemeister moves, shown in Figure 1. The number of such moves required to conn... |

1 |
7–23. 6 hass@math.ucdavis.edu tahl@math.biu.ac.il address
- Sem, Hamburg
- 1926
(Show Context)
Citation Context ...ructing a curve with the overcrossing arcs pushed slightly above the plane of the diagram and the remainder of the diagram lying in this plane. Alexander and Briggs [2] and independently Reidemeister =-=[11]-=- showed that two diagrams of the same knot can be connected through a sequence ∗ Supported in part by NSF grant DMS 3289292. 1of moves of three types, commonly called Reidemeister moves, shown in Fig... |