Bipyramid decompositions of multicrossing link complements (Q785706)

A multicrossing link projection allows for crossings of more than two strands at a time. The focus is on hyperbolic links, i.e., links whose complement carries a hyperbolic metric. These have been studied extensively by D. Thurston. Two decompositions of a link complement into hyperbolic bipyramids are developed, the face-centered decomposition and the crossing-centered decomposition. Both consist of bipyramids (such as octahedra in Thurston's 2-crossing projections), and they are dual in the sense that each bipyramid of one decomposition is itself decomposable into tetrahedra which, after regrouping and gluing, form the bipyramids of the other decomposition. The number of equatorial edges of a bipyramid is referred to by its size, and there is a relationship to the hyperbolic volume. Theorems 1 and 2 give formulae for these sizes depending on the crossing level data. Interestingly, these sizes can vary between adjacent crossing-centered bipyramids only by 4 if at all (Cor. 3). This leads to a description of the possible size sequences of crossing-centered bipyramid sizes of a crossing. The final section studies three types of weaves, which are \(\mathbb{Z}^2\) symmetric infinite link projections in \(\mathbb{R}^3\) giving rise to links in the \(3\)-sphere after dividing out the symmetry, and embedding the resulting thickened torus in a standard way into the \(3\)-sphere. Here, it turns out that the square weave, the triple weave and the right triangle weave are isometric to each other, with volume four times the octahedral volume (Theorem 8).