Minimal-transitivity finite tangles and catenated polyhedra

Tangles are defined as embeddings of connected graphs that contain knots and/or links as substructures. We enumerate and describe piecewise-linear embeddings of cubic and icosahedral tangles and linked polyhedra with transitivity [1 2]; that is, with one kind of vertex (all related by symmetry) and two kinds of edge. There are eight families of catenated polyhedra and ten families of tangles. For each family, we only describe the members with the largest girth, where girth is defined as the ratio of the shortest distance between edges with no shared vertices to the length of the longest edge.