Combinatorial cubings, cusps, and the dodecahedral knots. (English) Zbl 0773.57010

Topology ’90, Contrib. Res. Semester Low Dimensional Topol., Columbus/OH (USA) 1990, Ohio State Univ. Math. Res. Inst. Publ. 1, 17-26 (1992).
[For the entire collection see Zbl 0747.00024.]
Following H. S. M. Coxeter [Regular polytopes (1948; Zbl 0031.06502)] a plane regular \(p\)-gon is denoted by \(\{p\}\). A 3-dimensional polyhedron is called regular if its faces are \(\{p\}\)’s \(q\) surrounding each vertex. By Euler’s formula there are only 5 regular polyhedrons (Platonic solids): tetrahedron, octahedron, cube, icosahedron and dodecahedron denoted by \(\{3,3\}\), \(\{3,4\}\), \(\{4,3\}\), \(\{3,5\}\) and \(\{5,3\}\).
A 3-dimensional solid tessellation (or honeycomb) of a space form \((S^ 3,\mathbb{R}^ 3,\mathbb{H}^ 3)\) is a set of polyhedra fitting together to fill all space just once, so that every face of each polyhedron belongs to one other polyhedron. A solid tessellation is said to be regular if its cells are regular and equal. If these are \(\{p,q\}\)’s, and \(r\) of them surround an edge the tessellation is denoted by \(\{p,q,r\}\).
In the light of Thurston’s results regular tessellations of hyperbolic space by ideal Platonic solids are of special interest. By [H. S. M. Coxeter, Proc. Internat. Congr. Math. 1954, Amsterdam 3, 155-169 (1956; Zbl 0073.366)], there are only finitely many of those tessellations. Explicit examples of constant curvature finite volume 3-manifolds arising as a quotient from these possibilities are well known except for the tessellation \(\{5,3,6\}\). The authors introduce the dodecahedral knots \(D_ f\) and \(D_ s\) in \(S^ 3\) to fill this gap. Moreover, exactly four new knots in \(S^ 3\) are constructed, corresponding to the tessellations \(\{4,3,6\}\) and \(\{5,3.6\}\) of \(\mathbb{H}^ 3\), and united by a canonical construction from the Platonic solids.
Last but not least an infinite sequence \(K_ t\) of alternating fibered knots is established with the property that the complements of these new knots contain \(\pi_ 1\)-injective surfaces, which remain \(\pi_ 1\)-injective after ‘most’ Dehn surgeries. The closed 3-manifolds obtained by such surgeries are determined by their fundamental group, but are not known to be virtually Haken.
Reviewer: M.Heusener

MathOverflow Questions:

Quadratic cusp shape


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