Van Kampen diagrams, boundary labels, and diagram area for a presentation

Fix a presentation $G=⟨X\mid R⟩$ (Group presentation by generators and relations), with specified relator words on $X^{\pm 1}$ (Words in an alphabet with formal inverses, elementary cancellation, and reduced words). A van Kampen diagram is a nonempty finite connected simply connected combinatorial $2$-complex embedded in the plane. Here the $1$-skeleton is a finite graph, allowing loops and parallel edges, and each face is obtained by attaching a polygonal disc along a finite edge walk. Simply connected means that every closed path contracts to a point within the complex.

Each oriented edge is labelled by a letter of $X^{\pm 1}$; reversing its orientation inverts the letter. Reading the boundary of each face from a chosen corner gives a cyclic conjugate of a relator in $R$ or its inverse. No symmetrisation assumption on $R$ is required.

The outer boundary walk follows the unbounded complementary region, with the complex on its right. Choose a starting edge occurrence to obtain the boundary label, the literal word read along this walk. A bridge is traversed twice, once in each direction; at a cut vertex the walk follows the incident excursions in their planar cyclic order. Its length counts edge occurrences, not distinct edges. Reversing the walk gives the inverse word, and changing its starting occurrence gives a cyclic conjugate.

The one-vertex diagram has empty boundary word and area zero. Finite trees are also allowed: their outer walks traverse every edge twice. In particular, a single edge labelled $x$ has boundary word $x{x}^{-1}$, not the empty word, although these words are freely equivalent. Diagrams may also have bridges and cut vertices connecting faces. A diagram whose underlying space is a closed topological disc is called nonsingular; this is a special case, not a requirement on all diagrams.

The area is the number of $2$-cells, a natural number by The cardinality $\mid A\mid$ of a finite set. Boundary words are not freely reduced as part of this definition; any cancellation or spur insertion in a proof must account for its effect on the literal outer walk.