Singular planar labelled relator diagrams and their outer walks

Fix a presentation $\mathcal{P}=⟨X\mid R⟩$ (Group presentation by generators and relations) and words with formal inverses as in Words in an alphabet with formal inverses, elementary cancellation, and reduced words. A singular planar labelled relator diagram consists of an embedded finite connected plane multigraph, with loops and parallel edges as in Multigraphs, loops and directed graphs as variants distinct from the default finite simple graph, and finitely many characteristic polygon disks attached along cyclic edge-occurrence walks. Each characteristic map is injective on its open disk; its open-disk image is disjoint from the entire embedded graph, and the open-disk images of distinct faces are disjoint from one another. Edges are polygonal arcs, with distinct germs at the two ends of a loop. Geometric bends are not additional labelled edge occurrences.

Oriented edges carry letters in $X\bigsqcup X^{-1}$; reversing orientation inverts the letter. Each face's attaching walk reads a cyclic conjugate of a defining relator or its inverse. Occurrences are counted with multiplicity, even if an attaching walk repeats vertices or edges; neither its closed frontier nor its attaching map is required to be injective.

The outer boundary walk follows the edge occurrences bordering the unbounded region, in their plane cyclic order. An initial occurrence is specified when a literal linear word is read. Excursions at cut vertices are retained. A thin edge has no incident relator face. A bridge disconnects the graph when its open edge is deleted; a cut vertex disconnects it upon vertex deletion. These are distinct notions in the definition. The constructed diagrams below will have every thin edge a bridge traversed twice by the outer walk.

The diagram is contractible if its plane carrier has a contraction to a point. The single-vertex zero-face diagram is allowed. A Cayley vertex labelling is a map $g:V\to ⟨X\mid R⟩$ satisfying $g(w)=g(v)x$ on each oriented edge $v\to w$ labelled $x$. This condition includes loop edges.

The plane has its standard orientation. Whenever a literal outer word is read, the outer walk is the directed facial walk with the unbounded region locally on its left, and the specified initial occurrence is a directed edge occurrence. Thus both the traversal direction and the first letter are fixed, including for loop edges and the two occurrences of a bridge.