## The computational complexity of knot genus and spanning area

Venue: | electronic), arXiv: math.GT/0205057. MR MR2219001 |

Citations: | 6 - 0 self |

### BibTeX

@INPROCEEDINGS{Agol_thecomputational,

author = {Ian Agol and Joel Hass and William Thurston},

title = {The computational complexity of knot genus and spanning area},

booktitle = {electronic), arXiv: math.GT/0205057. MR MR2219001},

year = {}

}

### OpenURL

### Abstract

Abstract. We show that the problem of deciding whether a polygonal knot in a closed three-dimensional manifold bounds a surface of genus at most g is NP-complete. We also show that the problem of deciding whether a curve in a PL manifold bounds a surface of area less than a given constant C is NP-hard. 1.

### Citations

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Citation Context ...rally any tame knot, is equivalent to a knot that lies in the 1-skeleton of some triangulation. We formulate the problem of computing the genus as a language-recognition problem in the usual way; see =-=[2]-=-. In 1961 Schubert [21], in an extension of Haken’s work, showed the decidability of the problem: Problem. 3-MANIFOLD KNOT GENUS INSTANCE: A triangulated 3-dimensional manifold M,aknotKin the 1-skelet... |

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21 |
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Citation Context ...on to knots in R 3 or S 3 . (2) Casson has shown that a procedure to determine whether a 3-manifold is homeomorphic to the 3-sphere, following the 3-sphere recognition algorithm described in [18] and =-=[24]-=-, runs in time less than 3 t p(t), where p(t)isapolynomial.In the direction of lower bounds, it was shown in [13] that determining certain values of the Jones polynomial of alternating links is #P-har... |

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18 |
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Citation Context ...s equivalent to a knot that lies in the 1-skeleton of some triangulation. We formulate the problem of computing the genus as a language-recognition problem in the usual way; see [2]. In 1961 Schubert =-=[21]-=-, in an extension of Haken’s work, showed the decidability of the problem: Problem. 3-MANIFOLD KNOT GENUS INSTANCE: A triangulated 3-dimensional manifold M,aknotKin the 1-skeleton of M, and a natural ... |

13 |
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Citation Context ...hat each clause c ∈ C contains 3 literals. QUESTION: Is there a truth assignment for U such that each clause in C has exactly one true literal?s3826 IAN AGOL, JOEL HASS, AND WILLIAM THURSTON Schaefer =-=[19]-=- established that ONE-IN-THREE SAT is NP-complete. To prove Theorem 1, establishing that 3-MANIFOLD KNOT GENUS is NP-hard, we show that an arbitrary problem in ONE-IN-THREE SAT can be reduced in polyn... |

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1 | 109–123. Dept. of Mathematics, Statistics - Math - 1993 |