A Census of Slicings

Source abstract

A band is a closed connected set in the 2-sphere, bounded by one or more disjoint simple closed curves. Consider a band B with bounding curves J 1 , J 2 , … , J k . On each curve J i let there be chosen m i ≥ 0 points to be called vertices , with the restriction that the sum of the k integers m i is to be even. Write (1) Next consider a set of n disjoint open arcs in the interior of B which join the 2n vertices in pairs and partition the remainder of the interior of B into simply connected domains. We call the resulting dissection of B a slicing with respect to the given set of vertices. The arcs are the internal edges of the slicing and the simply connected domains are its internal faces , or slices .