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.


57N10 Topology of general \(3\)-manifolds (MSC2010)
57M50 General geometric structures on low-dimensional manifolds
57M25 Knots and links in the \(3\)-sphere (MSC2010)


Zbl 0773.57010


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