There are finitely many tessellations of 3-dimensional space-forms by regular Platonic solids. Explicit examples of constant curvature finite-volume 3-manifolds arising from these are well-known for all possibilities, except for the tessellation \( \{5,3,6\} \). We introduce the dodecahedral knots \( D_f \) and \( D_s \) in \( \mathbb{S}^3 \) to fill this gap. Techniques used illustrate the results on cusp structures and \( \pi_1 \)-injective surfaces of alternating link complements obtained by Aitchison, Lumsden and Rubinstein [1991].
The Borromean rings and figure-eight knot arise from the tessellation of hyperbolic 3-space by regular ideal octahedra and tetrahedra respectively. We produce exactly four new links in \( \mathbb{S}^3 \), 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.
The dodecahedral knot \( D_f \) is the third in an infinite sequence of fibred, alternating knots, the first member of which being the figure-eight. The complements of these new links 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 groups, but are not known to be virtually Haken.
