From Edge Pairings to Boundary Words
A polygon quotient is described by specifying which boundary edges are identified and how points on each paired pair correspond. A boundary word is a compact record of that information: each paired edge receives the same letter, and the sign records its direction relative to a chosen reference. The sign is essential. It tells us whether boundary parameters on the paired edges match in the same order or in reverse.
Let \(P\) be a polygonal disk with its boundary edges read in order around the boundary, in one chosen direction. Call the directed edges \(e_1,\dots,e_n\). Give each edge a boundary parameter \(t_i\in[0,1]\), with \(t_i=0\) at its starting vertex and \(t_i=1\) at its ending vertex in that traversal. We consider pairings in which each edge is paired with exactly one other edge, and the identification is the linear map that matches or reverses these parameters.
The reference direction is a bookkeeping choice, not an extra identification. If the two occurrences have the same sign, their boundary parameters match directly. If they have opposite signs, one boundary parameter must be reversed. In both cases, points with the same coordinate in the chosen reference direction are identified.
Proof. Fix one paired pair, with edges \(e_i\) and \(e_j\), and let \(r_i,r_j\in[0,1]\) denote their parameters measured in the chosen reference direction for their label. If an occurrence has a positive sign, its boundary parameter agrees with its reference parameter, so \(r_k=t_k\). If it has a negative sign, the two directions disagree, so \(r_k=1-t_k\). The pairing identifies points with equal reference parameter, which means \(r_i=r_j\). If both signs are positive, this equation gives \(t_i=t_j\); if both are negative, it gives \(1-t_i=1-t_j\), again giving \(t_i=t_j\). If the signs are opposite, the equation gives either \(t_i=1-t_j\) or \(1-t_i=t_j\), both equivalent to \(t_j=1-t_i\). Conversely, if the identification has \(t_j=t_i\), the two occurrences must have the same sign under the reference-direction convention; if it has \(t_j=1-t_i\), they must have opposite signs. Applying this argument to every pair proves the assertion. \(\square\)
Reading and Writing the Signs
To write a word from a pairing, first traverse the boundary and record its edges in order. Then handle each pair separately: choose a reference direction on one of its edges, compare both boundary directions with that reference, and put the same letter on both edges with the appropriate signs. A useful check is that every label appears exactly twice. The word records the cyclic order and direction data; it does not, by itself, compute which polygon vertices become equivalent in the quotient.
Worked Example: A Square with Two Reversed Parameter Matches
Let the square have boundary edges \(e_1,e_2,e_3,e_4\), read consecutively around the boundary. Pair \(e_1\) with \(e_3\), and \(e_2\) with \(e_4\). For each pair, choose the boundary direction on the first listed edge as the reference direction, and identify points with the same reference parameter.
On \(e_1\), the boundary direction is the reference direction, so its letter is \(a\). On \(e_3\), the boundary direction is opposite to the reference direction, so its letter is \(a^{-1}\). The same comparison for \(e_2,e_4\) gives \(b,b^{-1}\). Reading from \(e_1\), the word is \(a\,b\,a^{-1}\,b^{-1}\).
Let \(t_i\) be the boundary parameter on \(e_i\). For the \(a\)-pair, the signs are opposite, so \(t_3=1-t_1\). For the \(b\)-pair, they are also opposite, so \(t_4=1-t_2\). For example, \(t_1=0\) is identified with \(t_3=1\), and \(t_1=1\) with \(t_3=0\). Likewise, \(t_2=0\) matches \(t_4=1\), and \(t_2=1\) matches \(t_4=0\).
Check: At the midpoint of either edge, the parameter is \(1/2\), and reversing it still gives \(1-1/2=1/2\). Thus the midpoint matches the midpoint for both pairs. Each of \(a\) and \(b\) occurs exactly twice, and the opposite signs agree with the reversed parameter formulas.
Worked Example: Same Signs Mean Direct Parameter Matching
Take a hexagon with boundary edges \(e_1,\dots,e_6\), and pair \(e_1\) with \(e_2\), \(e_3\) with \(e_4\), and \(e_5\) with \(e_6\). Choose the boundary direction of the first edge in each pair as its reference direction. Suppose the first pair is identified by matching equal boundary parameters, while the second and third pairs are identified by reversing boundary parameters.
For \(e_1,e_2\), both boundary directions agree with the reference, so the two signs are positive: \(a,a\). For \(e_3,e_4\), the identification reverses the parameters, so the signs are opposite: \(b^{-1},b\). For \(e_5,e_6\), they are also opposite: \(c,c^{-1}\). The resulting word is \(a\,a\,b^{-1}\,b\,c\,c^{-1}\).
Decoding gives \(t_2=t_1\) for the \(a\)-pair, \(t_4=1-t_3\) for the \(b\)-pair, and \(t_6=1-t_5\) for the \(c\)-pair. For the first pair, \(t_1=0\) matches \(t_2=0\), and \(t_1=1\) matches \(t_2=1\). For the second, \(t_3=0\) matches \(t_4=1\), and \(t_3=1\) matches \(t_4=0\). The same endpoint checks for the third pair give \(t_5=0\) matched to \(t_6=1\), and \(t_5=1\) matched to \(t_6=0\).
Check: The word has six letters, accounting for the six boundary edges, and each of \(a,b,c\) occurs twice. Only the \(a\)-pair has equal signs, so only that pair matches boundary parameters directly.
Worked Example: Decoding a Mixed Word
Consider the eight-edge word \(p^{-1}q\,r^{-1}p\,q^{-1}r\,s\,s\), read in the displayed order. Each label occurs twice, so the letters specify four pairs. For each label compare its two signs.
The \(p\)-occurrences are \(p^{-1}\) and \(p\), so their signs are opposite and the parameters satisfy \(t_4=1-t_1\). The \(q\)-occurrences are \(q\) and \(q^{-1}\), so \(t_5=1-t_2\). The \(r\)-occurrences are \(r^{-1}\) and \(r\), so \(t_6=1-t_3\). The two \(s\)-occurrences are both positive, so \(t_8=t_7\).
For example, the \(p\)-identification sends the point with \(t_1=0\) to the point with \(t_4=1\), while \(t_1=1\) matches \(t_4=0\). The \(s\)-identification sends \(t_7=0\) to \(t_8=0\) and \(t_7=1\) to \(t_8=1\). The \(q\)- and \(r\)-pairs follow their displayed formulas in the same way.
Check: The word contains eight occurrences and four labels, each appearing exactly twice. Three pairs have opposite signs and therefore reverse boundary parameters; the \(s\)-pair has equal signs and therefore matches them directly.
What the Word Records—and What Can Change
The signed word is a concise encoding of the edge-pairing data under the convention above. It records which edges are paired, their order along the boundary, and whether each pairing matches or reverses the boundary parameters. It does not specify the shape of the polygon, and the same combinatorial word can be drawn on polygons with different geometric shapes, provided the edge pairings use the stated linear parameter rules.
Changing the reference direction for one label changes both of its signs. For example, the word \(a\,b\,a^{-1}\,b^{-1}\) becomes \(a^{-1}\,b\,a\,b^{-1}\) if the reference for \(a\) is reversed. The two occurrences of \(a\) still have opposite signs, so the identification of that edge pair is unchanged.
Proof. Reversing the reference direction interchanges the two possible comparisons between a boundary direction and the reference: an agreeing direction becomes disagreeing, and a disagreeing direction becomes agreeing. Thus both signs for that label flip. If the signs were equal before the change, they remain equal; if they were opposite before, they remain opposite. By the decoding theorem, the parameter rule for the pair is unchanged in either case. \(\square\)
Changing the starting edge used to write the word only cyclically shifts its letters. Reversing the direction in which the boundary is read reverses the order of the occurrences and flips every sign: each edge is then traversed in the direction opposite to the original boundary traversal. These changes affect the written presentation, not the pairing itself, as long as the reference directions are interpreted consistently.
A Common Sign Error
A frequent mistake is to read \(a^{-1}\) as though it named a different edge from \(a\), or to ignore the exponent and assume that every pair matches equal boundary parameters. In the square example, the pair \(a,a^{-1}\) does not satisfy \(t_3=t_1\). Its signs are opposite, so the correct rule is \(t_3=1-t_1\). For instance, \(t_1=0\) matches \(t_3=1\), not \(t_3=0\). Writing \(t_3=t_1\) would therefore give the wrong endpoint matching. The inverse symbol is a direction marker for the same edge label, not a second label or an instruction to omit the pairing.
Before using a boundary word, check three things: every edge occurrence has a sign; every label occurs exactly twice; and the sign comparison for each pair agrees with the intended parameter rule. These checks verify the encoded edge data without requiring any conclusions about the quotient’s vertex classes or its local surface structure.
Check Your Understanding
For each question, use the convention that equal signs give direct parameter matching and opposite signs give reversed parameter matching.
- Decode the word \(a\,b\,a\,b^{-1}\). Which pairs match boundary parameters directly?
- A pair of edges is identified by \(t_j=1-t_i\). What sign relationship must its two occurrences have?
- Write the signed word for four consecutively traversed edges paired as \((e_1,e_3)\) and \((e_2,e_4)\), if both pairings match equal boundary parameters and the first edge of each pair sets the reference direction.
- Why does reversing the reference direction for a label leave its edge identification unchanged?
- What is wrong with a proposed word in which one label occurs three times and another occurs only once?