Sc toolkit labelled planar disc diagram

Over the symmetrised presentation of Sc toolkit symmetrised relators and pieces, a diagram is a finite connected simply connected combinatorial $2$-complex embedded in the plane. Its $1$-skeleton is a finite graph, allowing loops and parallel edges. Each oriented edge has a letter of $X^{\pm 1}$ as label; reversal inverts that letter, as in Words in an alphabet with formal inverses, elementary cancellation, and reduced words. Faces are polygonal $2$-cells attached along finite edge walks, each reading a member of $R$. Orienting a face oppositely or changing its starting corner is allowed by symmetrisation. Area is the number of faces.

The outer boundary walk follows the unbounded complementary region, with the complex on its right. Choose a starting edge occurrence to obtain a word. A bridge is traversed twice, once in each direction. Length counts occurrences, not distinct edges. At a cut vertex the walk completes the incident excursions in their planar order. The diagram consisting of one vertex has empty boundary and area zero.

A nonsingular disc diagram has underlying space a closed topological disc. General diagrams may have cut vertices, bridges and spurs (vertices incident with exactly one edge germ). The topological frontier is a set; the outer walk is a parametrised walk and must not be confused with that set. A finite tree is a zero-face diagram.