## 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:

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