Symmetric colorings of polypolyhedra

Polypolyhedra (after R. Lang) are compounds of edge-transitive 1-skeleta. There are 54 topologically different polypolyhedra, and each has icosidodecahedral, cuboctahedral, or tetrahedral symmetry, all are realizable as modular origami models with one module per skeleton edge. Consider a coloring in which each edge of a given component receives a different color, and where the coloring (up to global color permutation) is invariant under the polypolyhedron's symmetry group. On the Five Intersecting Tetrahedra, the edges of each color form visual bands on the model, and correspond to matchings on the dodecahedron graph. We count the number of such colorings and give three proofs. For each of the non-polygon-component polypolyhedra, there is a corresponding matching coloring, and we count the number of these matching colorings. For some of the non-polygon-component polypolyhedra, there is a corresponding visual-band coloring, and we count the number of these band colorings

Aperiodic colorings and tilings of Coxeter groups

We construct a limit aperiodic coloring of hyperbolic groups. Also we construct limit strongly aperiodic strictly balanced tilings of the Davis complex for all Coxeter groups

Coloring Link Diagrams And Conway-type Polynomial Of Braids

In this paper we define and present a simple combinatorial formula for a 3-variable Laurent polynomial invariant of conjugacy classes in Artin braid group $B_m$. We show that this Laurent polynomial satisfies the Conway skein relation and its coefficients are Vassiliev invariants of braids.Comment: This is a revised version. To appear in Topology and its Application