## Algorithms for recognizing knots and 3-manifolds (1998)

Venue: | Chaos, Solitons and Fractals |

Citations: | 6 - 3 self |

### BibTeX

@ARTICLE{Hass98algorithmsfor,

author = {Joel Hass},

title = {Algorithms for recognizing knots and 3-manifolds},

journal = {Chaos, Solitons and Fractals},

year = {1998},

volume = {9},

pages = {569--581}

}

### OpenURL

### Abstract

Algorithms are of interest to geometric topologists for two reasons. First, they have bearing on the decidability of a problem. Certain topological questions, such as finding a classification of four dimensional manifolds, admit no solution.

### Citations

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Introduction to Automata Theory, Languages, and Computation
- Hopcroft, UlIman
- 1979
(Show Context)
Citation Context ...e implications to a real world implementation of an algorithm, but is independent of the decidability of a question. A good reference for the notions of Turing machine, algorithms and decidability is =-=[8]-=-. There are natural questions that do not admit any algorithm to decide them. A famous example is the word problem for finitely presented groups. Given a group described by a collection of generators ... |

111 |
An introduction to the theory of groups
- Rotman
- 1994
(Show Context)
Citation Context ...en product of generators represents the identity element. It was shown by Novikov and Boone that there are groups in which there is no algorithm to decide this question. An exposition can be found in =-=[20]-=-. Closely related is the question of whether a finitely presented group G is isomorphic to the trivial group, which also cannot be decided by an algorithm. Note however that it is easy to construct an... |

52 |
Theorie der Normalflächen
- Haken
- 1961
(Show Context)
Citation Context ... To give algorithms to solve problems in 3-dimensional topology. We will concentrate on the second contribution, and describe Haken’s algorithm to recognize the unknot among the knots in the 3-sphere =-=[2]-=-. A knot is the unknot if it bounds an embedded disk in S 3 . The algorithm will search for this disk, and either produce it or show it does not exist in a finite amount of time. In fact, the same alg... |

50 |
Embedded minimal surfaces, exotic spheres, and manifolds with positive Ricci curvature
- Meeks, Yau
- 1982
(Show Context)
Citation Context ...class. One successful idea, developed originally by Meeks and Yau, is to put a Riemannian metric on the 3-manifold and find a surface of least area in the homotopy or isotopy class of the surface [17]=-=[18]-=-. It is a non-trivial result that a least area surface tends to minimize its self-intersections, as well as being rather rigidly situated 3sin the 3-manifold. In Thurston’s development of the theory o... |

47 |
Algorithms for the complete decomposition of a closed 3
- Jaco, Tollefson
- 1995
(Show Context)
Citation Context ... difficult, cannot occur in a knot complement. In each case we can obtain a contradiction. An argument used in [9] for closed surfaces, extends to our setting. This extension is explicitly derived in =-=[11]-=-. In each of the above cases, if C = A + B then we can find A ′ and B ′ such that C = A ′ + B ′ where A ′ is an essential disk and B ′ is a torus or an annulus. The essential normal disk A ′ has lower... |

43 | Polyhedral minimal surfaces, Heegaard splittings and decision problems for 3–dimensional manifolds, from: “Geometric topology - Rubinstein - 1993 |

32 |
Residual finiteness for 3-manifolds, in: Combinatorial Group Theory and Topology
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(Show Context)
Citation Context ...not is not the unknot, then it is a consequence of the existence of geometric structures on knot complements that the fundamental group of the knot complement is a non-abelian residually finite group =-=[7]-=-. Residually finite means that for any non-trivial element of the group, there is a homomorphism to some finite group which takes that element to a non-trivial element of the finite group. There is no... |

32 |
The isomorphism problem for hyperbolic groups
- Sela
- 1995
(Show Context)
Citation Context ... have recently announced an algorithm to recognize Seifert Fibered manifolds. Sela gave an algorithm to decide whether two 3-manifolds with Gromov hyperbolic fundamental group are homotopy equivalent =-=[24]-=-. Birman and Hirsch [1] have recently announced a new algorithm, based on work of Birman and Menasco, which detects whether a knot presented in braid form is the trivial knot. Jaco and Tollefson descr... |

32 |
Almost normal surfaces in 3-manifolds
- Stocking
- 2000
(Show Context)
Citation Context ...ant argument that this algorithm works by using the notion of thin position [26]. Rubinstein also described algorithms to recognize Lens spaces and other 3-manifolds of small genus. See also Stocking =-=[25]-=-. Rubinstein and Rannard have recently announced an algorithm to recognize Seifert Fibered manifolds. Sela gave an algorithm to decide whether two 3-manifolds with Gromov hyperbolic fundamental group ... |

29 |
Thin position and the recognition problem for
- Thompson
- 1994
(Show Context)
Citation Context ...this to describe an algorithm which decides whether a 3-manifold is homeomorphic to the 3-sphere [22]. Thompson gave an elegant argument that this algorithm works by using the notion of thin position =-=[26]-=-. Rubinstein also described algorithms to recognize Lens spaces and other 3-manifolds of small genus. See also Stocking [25]. Rubinstein and Rannard have recently announced an algorithm to recognize S... |

28 |
Insolubility of the problem of homeomorphy
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(Show Context)
Citation Context ... been classified for some time, a classification remains elusive for 3-manifolds. Markov showed that the Novikov-Boone results implied that the classification problem for 4-manifolds was not solvable =-=[15]-=-. Given a finitely presented group, a compact 4-manifold (or n-manifold with n > 4) can be constructed with that group as its fundamental group. Markov showed that a classification of 4-manifolds coul... |

23 |
The classification of knots and 3-dimensional spaces
- Hemion
- 1992
(Show Context)
Citation Context ...decide not just whether a knot is the unknot, but whether two arbitrary knots K1 and K2 are the same. This was carried out by Haken and Hemion using some additional arguments based on normal surfaces =-=[5]-=-. We outline very briefly here an extension of the algebraic algorithm described above that can also give such a classification. This emerges from work of Thurston, and was also related to me by Kuper... |

22 |
Topology of three–dimensional manifolds and the embedding problems in minimal surface theory
- Meeks, Yau
- 1980
(Show Context)
Citation Context ...opy class. One successful idea, developed originally by Meeks and Yau, is to put a Riemannian metric on the 3-manifold and find a surface of least area in the homotopy or isotopy class of the surface =-=[17]-=-[18]. It is a non-trivial result that a least area surface tends to minimize its self-intersections, as well as being rather rigidly situated 3sin the 3-manifold. In Thurston’s development of the theo... |

19 |
Affine structure on 3-manifolds
- Moise
- 1952
(Show Context)
Citation Context ...m arises for 3-manifolds, it is still unknown whether they can be classified. 2sHere is an example of something that is not a classification of 3-manifolds. All closed 3-manifolds can be triangulated =-=[16]-=-. Since there are only finitely many ways to glue together k tetrahedra, we can construct all of these systematically. A resulting complex is a manifold exactly when the links of all vertices are 2-sp... |

18 |
Bestimmung der Primfaktorzerlegung von Verkettungen
- Schubert
- 1961
(Show Context)
Citation Context ...there are only finitely many. Some subset of these fundamental solutions, those that also satisfy the quadrilateral condition, correspond to embedded surfaces. The following result is due to Schubert =-=[23]-=-. Lemma 7 If C is a connected normal surface in an irreducible 3-manifold and C is not fundamental then we can find connected normal surfaces A and B so that C = A + B. If A and B are chosen to minimi... |

15 | Recent results on sufficiently large 3–manifolds - Waldhausen - 1997 |

14 |
The computational complexity of knot and link problems, preprint
- Hass, Lagarias, et al.
- 1998
(Show Context)
Citation Context ...so develop the important idea of a vertex surface, a type of fundamental normal surface introduced by Jaco and Oertel in [9], which gives a more specialized representative for a class of surfaces. In =-=[4]-=- a bound for the complexity of the unknotting algorithm is given. It is also shown that this problem is in the class NP. Casson has recently announced a computation of the compexity of th 3-sphere rec... |

9 |
Connections between topological and group theoretical decision problems
- Haken
- 1973
(Show Context)
Citation Context ...ser’s Theorem led to the establishment by Milnor of a unique factorization theorem for 3-manifolds into prime pieces. See [6] for an exposition. An almost identical argument proves a theorem of Haken =-=[3]-=-. Theorem 4 Let M be a triangulated 3-manifold with t 3-simplices and let k(M) = dim(H1(M; Z2)) + dim(H1(M; Z)) + 6t. Then M contains at most k(M) disjoint, non-parallel, incompressible surfaces. 5 Re... |

7 |
Recognizing the unknot, preprint
- Birman
- 1997
(Show Context)
Citation Context ... an algorithm to recognize Seifert Fibered manifolds. Sela gave an algorithm to decide whether two 3-manifolds with Gromov hyperbolic fundamental group are homotopy equivalent [24]. Birman and Hirsch =-=[1]-=- have recently announced a new algorithm, based on work of Birman and Menasco, which detects whether a knot presented in braid form is the trivial knot. Jaco and Tollefson describe algorithms to const... |

7 | Some problems on 3-manifolds - Waldhausen - 1978 |

4 |
An Algorithm to Decide If a 3-Manifold Is a
- Jaco, Oertel
- 1984
(Show Context)
Citation Context ...oes not bound a disk on the torus. The following lemma allows us to reduce the search for an unknotting disk to the finite collection of fundamental surfaces. Our proof is based on that of Jaco-Oertel=-=[9]-=-. Lemma 8 Suppose C is a normal disk in a knot complement, with minimal weight among all normal disks with essential boundary. Then C is fundamental. 11sProof: If the lemma fails, then we can find two... |

3 |
A piecewise linear theory of minimal surfaces in 3-manifolds
- Jaco, Rubinstein
- 1988
(Show Context)
Citation Context ...l disk A ′ has lower weight than C, a contradiction. Note: An alternate way to obtain a contradiction can be found by taking C to be of minimal complexity, as measured in the sense of Jaco-Rubinstein =-=[10]-=-. In this setting the complexity measures the length of intersection of a normal surface with the 2-skeleton of a triangulation. Theorem 9 There is an algorithm to decide whether a knot is the unknot.... |

3 | Algorithms for the recognition of the three-dimensional sphere (after - Matveev - 1995 |

1 | Classification problems in low-dimensional topology, Geometric and algebraic topology - Johannson - 1986 |

1 |
Geschlossene Flachen in dreidimesionalen Mannifgfaltigkeiten, Jahresericht der Ent
- Kneser
- 1929
(Show Context)
Citation Context ...rises whether it’s possible to keep on splitting M indefinitely in a non-trivial way, into more and more pieces. Normal surfaces were introduced by Kneser, who used them to prove the following result =-=[13]-=-. Theorem 1 Let M be a triangulated 3-manifold with t 3-simplices and let k(M) = dim(H1(M; Z2)) + dim(H1(M; Z)) + 6t. Them M can be decomposed non-trivially along 2-spheres into at most k(M) pieces. S... |

1 |
On Dehn's lemma and the asphericity of knots
- Papakyriakopoulous
- 1957
(Show Context)
Citation Context ...undary a collection of disjoint simple closed curves. The boundary of T is a 2-sphere, and each simple closed curve in F ∩∂T cuts ∂T into two disks. By applying the Loop Theorem of Papakyriakopoulous =-=[19]-=- [6], we can find a series of compressions in T which yield a new surface, all of whose components meet T in disks and 2-spheres. Since T is a ball, and therefore irreducible, the 2-sphere components ... |

1 |
An algorithm to recognize the 3-sphere, to appear
- Rubinstein
- 1994
(Show Context)
Citation Context ...]. Rubinstein extended the notion of normal surfaces to the concept of almost normal surfaces. He used this to describe an algorithm which decides whether a 3-manifold is homeomorphic to the 3-sphere =-=[22]-=-. Thompson gave an elegant argument that this algorithm works by using the notion of thin position [26]. Rubinstein also described algorithms to recognize Lens spaces and other 3-manifolds of small ge... |

1 | The geometry and topology of 3-dimensional manifolds - Thurston - 1978 |