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.