A census of hyperbolic Platonic manifolds and augmented knotted trivalent graphs
نویسندگان
چکیده
We call a 3-manifold Platonic if it can be decomposed into isometric Platonic solids. Generalizing an earlier publication by the author and others where this was done in case of the hyperbolic ideal tetrahedron, we give a census of hyperbolic Platonic manifolds and all of their Platonic tessellations. For the octahedral case, we also identify which manifolds are complements of an augmented knotted trivalent graph and give the corresponding link. A (small version of) the Platonic census and the related improved algorithms have been incorporated into SnapPy. The census also comes in Regina format.
منابع مشابه
The Volume Conjecture for Augmented Knotted Trivalent Graphs
We propose to generalize the volume conjecture to knotted trivalent graphs and we prove the conjecture for all augmented knotted trivalent graphs. As a corollary we find that for any link L there is a link containing L for which the volume conjecture holds.
متن کاملThe Algebra of Knotted Trivalent Graphs and Turaev’s Shadow World
Knotted trivalent graphs (KTGs) form a rich algebra with a few simple operations: connected sum, unzip, and bubbling. With these operations, KTGs are generated by the unknotted tetrahedron and Möbius strips. Many previously known representations of knots, including knot diagrams and non-associative tangles, can be turned into KTG presentations in a natural way. Often two sequences of KTG operat...
متن کاملHomomorphic Expansions for Knotted Trivalent Graphs — Fixing an Anomaly
It had been known since old times ([MO], [Da]) that there exists a universal finite type invariant (“an expansion”) Z for Knotted Trivalent Graphs (KTGs), and that it can be chosen to intertwine between some of the standard operations on KTGs and their chord-diagrammatic counterparts (so that relative to those operations, it is “homomorphic”). Yet perhaps the most important operation on KTGs is...
متن کاملHomomorphic Expansions for Knotted Trivalent Graphs
It had been known since old times [MO, CL, Da] that there exists a universal finite type invariant (“an expansion”) Z for Knotted Trivalent Graphs (KTGs), and that it can be chosen to intertwine between some of the standard operations on KTGs and their chord-diagrammatic counterparts (so that relative to those operations, it is “homomorphic”). Yet perhaps the most important operation on KTGs is...
متن کامل3-manifolds from Platonic solids
The problem of classifying, up to isometry, the orientable spherical and hyperbolic 3-manifolds that arise by identifying the faces of a Platonic solid is formulated in the language of Coxeter groups. This allows us to complete the classification begun by Best [Canad. J. Math. 23 (1971) 451], Lorimer [Pacific J. Math. 156 (1992) 329], Richardson and Rubinstein [Hyperbolic manifolds from a regul...
متن کامل