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**Geodesic surfaces in knot complements.**
*(English)*
Zbl 0891.57017

Alan Reid has shown by arithmetic arguments that the complement of the figure-eight knot in \(S^3\), with its complete hyperbolic metric, contains infinitely many commensurability classes of totally geodesic immersed closed surfaces. According to Reid, the figure-eight knot is the only arithmetic knot. No other hyperbolic knot complements were known to contain immersed totally geodesic closed surfaces. The aim of the paper is to investigate geometrical properties of the dodecahedral knots \(D_f\) and \(D_s\) discovered earlier by the authors [ Math. Res. Inst. Publ. 1, 17-26 (1992; Zbl 0773.57010)]. \(D_f\) and \(D_s\) are the only known nonarithmetic knots with hidden symmetries. (That is, the knot complements nonnormally cover some orbifold.) First of all, direct geometrical arguments are given for the existence of closed totally geodesic immersed surfaces in these knot complements. Then two pictures of the view from infinity of an ideal dodecahedron are drawn which enable us to see directly the cusp and trace fields of the above knots. Finally, a remarkable empirical fact is obtained using the Snap Pea program by J. Weeks that the spectrum for \(D_s\) appears as a subset of the spectrum of \(D_f\). At this moment no reasonable explanation of this fact is known.

Reviewer: A.D.Mednykh (Novosibirsk)

### MSC:

57N10 | Topology of general \(3\)-manifolds (MSC2010) |

57M50 | General geometric structures on low-dimensional manifolds |

57M25 | Knots and links in the \(3\)-sphere (MSC2010) |

### Keywords:

immersed and embedded totally geodesic closed surfaces; hyperbolic 3-manifold; cusp and trace fields; length spectra### Citations:

Zbl 0773.57010### Software:

SnapPea
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\textit{I. R. Aitchison} and \textit{J. H. Rubinstein}, Exp. Math. 6, No. 2, 137--150 (1997; Zbl 0891.57017)

### References:

[1] | Aitchison I. R., Topology ’90 pp 17– (1992) |

[2] | Coxeter H. S. M., Regular Polytopes (1948) |

[3] | Coxeter, H. S. M. ”Regular honeycombs in hyperbolic space”. Proceedings of the International Congress of Mathematicians. vol. Ill, pp.155–169. Groningen: Erven P. Noordhoff N.V. [Coxeter 1956], (Amsterdam, 1954) · Zbl 0056.38603 |

[4] | Hass J., Topology 31 (3) pp 493– (1992) · Zbl 0771.57007 |

[5] | Lee Y. W., Trans. Amer. Math. Soc. 290 (2) pp 735– (1985) |

[6] | Lee Y. W., Trans. Amer. Math. Soc. 290 (2) pp 735– (1985) |

[7] | Long D. D., Bull. London Math. Soc. 19 (5) pp 481– (1987) · Zbl 0596.57011 |

[8] | Maclachlan C., Low-dimensional topology and Kleinian groups pp 305– (1986) |

[9] | Maclachlan C., Math. Proc. Cambridge Philos. Soc. 102 (2) pp 251– (1987) · Zbl 0632.30043 |

[10] | Menasco W., Topology ’90 pp 215– (1992) |

[11] | Murasugi K., Amer. J. Math. 85 pp 544– (1963) · Zbl 0117.17201 |

[12] | Neumann W. D., Topology ’90 pp 273– (1992) |

[13] | Reid A. W., J. London Math. Soc. (2) 43 (1) pp 171– (1991) · Zbl 0847.57013 |

[14] | Reid A. W., Proc. Edinburgh Math. Soc. (2) 34 (1) pp 77– (1991) · Zbl 0714.57010 |

[15] | Riley R. F., Computers in geometry and topology pp 297– (1989) |

[16] | Weeks J. R., ”SnapPea – (software for the study of hyperbolic three-manifolds)” |

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