This paper classifies when conjugating permutations in transitive pairs preserves transitivity, explores exceptional cases, and relates these to graph deletion problems, with implications for understanding dessins d'enfants and Galois actions.
Contribution
It provides a complete classification of transpositions that preserve transitivity in permutation pairs and connects these to graph deletion and topological surface genus changes.
Findings
01
Identified exactly which transpositions preserve transitivity.
02
Characterized exceptional cases and their properties.
03
Linked permutation conjugation to graph deletion problems.
Abstract
Let E be a finite set. Given permutations x and y of E that together generate a transitive subgroup, for which s is it true that x and the conjugate of y by s also generate a transitive subgroup? Such transitive permutation pairs encode dessins d'enfants, important graph-theoretic objects which are also known to have great arithmetic significance. The absolute Galois group acts on dessins d'enfants and permutes them in a very mysterious way. Two dessins d'enfants that share certain elementary combinatorial features are related by conjugations as above, and dessins d'enfants in the same Galois-orbit share these features and more, so it seems worthwhile to have a good answer to the above question. I classify, relative to x and y, exactly those transpositions s for which the new pair is guaranteed to be transitive. I also provide examples of the "exceptional" s which show the range ofā¦
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Taxonomy
TopicsAlgebraic Geometry and Number Theory Ā· Finite Group Theory Research Ā· Coding theory and cryptography
Let E be a finite set and SEā its symmetric group. Given Ļ,Ļā²āSEā that together generate a transitive subgroup, for which sāSEā is it true that sĻsā1,Ļā² also generate a transitive subgroup? Such transitive permutation pairs encode dessin dāenfants, important graph-theoretic objects which are also known to have great arithmetic significance. The absolute Galois group Gal(Qā/Q) acts on dessins dāenfants and permutes them in a very mysterious way. Two dessins dāenfants that share certain elementary combinatorial features are related by conjugations as above, and dessins dāenfants in the same Gal(Qā/Q)-orbit share these features and more, so it seems worthwhile to have a good answer to the above question. I classify, relative to Ļ,Ļā², exactly those transpositions s for which the new pair is guaranteed to be transitive. I also provide examples of the āexceptionalā s which show the range of possible behavior and prove that the above question for the exceptional cases is equivalent to a natural question about deletion in graphs that may have a good answer in this more structured world of topological graphs. Finally, I classify transpositions s according to how they change the genus of the surface underlying the dessin dāenfant of Ļ,Ļā². Some of the tools, like the Reroute Operation/Theorem, may have use beyond Dessins dāEnfants.
Key words and phrases:
dessin dāenfant, non-simultaneous conjugation, transitive subgroup, graph surgery
The field of Rational Numbers is denoted Q and the field of Complex Numbers is denoted C. The algebraic closure of Q in C is denoted Qā. The compactification of C, the Riemann Sphere, is denoted C. The (topological) sphere is denoted S and the (topological) torus T. For E a finite set, ā£E⣠denotes its cardinality and SEā denotes its symmetric group. For ĻāSEā, an Ļ-orbit is an orbit under the action on E by the cyclic subgroup generated by Ļ, and O(Ļ) denotes the total number of Ļ-orbits. For ĻāSEā, a Ļ-cycle is simply a cycle in the disjoint cycle decomposition of Ļ. For Ļ1ā,Ļ2āāSEā, the pair (Ļ1ā,Ļ2ā) is transitive iff the subgroup of SEā generated by Ļ1ā,Ļ2ā is transitive, i.e. iff for any a,bāE there is a word w in Ļ1ā,Ļ2ā such that w(a)=b. For Ļ,sāSEā, abbreviate Ļs=defsā Ļā sā1.
Motivation
The key reason why transitive permutation pairs are important is that they encode, in a way that facilitates proof and computation, dessins dāenfants, which are important objects in both Number Theory and Topological Graph Theory. Related concepts are cellular embeddings, maps, rotation systems, ribbon graphs, etc.
A dessin dāenfant is a triple (D,X,ι) consisting of a finite bicolored111āBicoloredā means that one of two colors is assigned to each vertex and every edge is incident to one vertex of each color. This differs from the notion of ābipartiteā only in that a choice of color for each vertex is fixed. graph D, a connected oriented compact surface X without boundary, and an embedding ι:DāŖX such that the complement Xāι(D) is homeomorphic to a union of open discs. For the last requirement, it is necessary but not sufficient that D is a connected graph. The symbols ā and ā will be used to indicate the colors of the vertices, and a vertex will be referred to as either a ā-vertex or a ā-vertex accordingly. Usually the embedding ι will be omitted from the notation.
These objects were known, in various slightly different forms, for quite a long time. A newer reason to consider such an object can be found in the following theorem of BelyÄ [belyi] and others: A compact Riemann surface S is defined over Qā if and only if there is a holomorphic function f:SāC ramified over at most three values. If (S,f) is such a BelyÄ pair, with the ramification values normalized to be 0,1,āāC, then a dessin dāenfant (D,X) is obtained as follows: define X to be the mere topological surface of S and define D to be the preimage in X under f of [0,1], where preimages of [math] are colored by ā, preimages of 1 are colored by ā, and preimages of (0,1) are edges. Conversely, given a dessin dāenfant, one may construct a BelyÄ pair (cf. §4.2 of [GG]). These two constructions yield, modulo certain natural equivalence relations on each side, a bijection. The number-theoretic significance of this is due to an observation by Grothendieck [sketch]: The absolute Galois group Gal(Qā/Q) permutes, very mysteriously, the set of dessins dāenfants (D,X) via its permutation of the equivalent objects (S,f). It is perhaps worth emphasizing that a dessin dāenfant is, superficially, a purely topological object but endows its surface, in particular, with a complex structure. To appreciate this, consider the diversity of complex structures, the elliptic curves, on T.
I now describe the role played by permutations. Suppose (D,X) is a dessin dāenfant, and denote by E the set of edges of the graph D. The orientation of X cyclically orders the edges incident to each ā-vertex, which defines a disjoint cycle decomposition, i.e. a permutation ĻāāāSEā. Similarly, (D,X) defines a permutation ĻāāāSEā. The pair (Ļāā,Ļāā) is called the monodromy pair of (D,X), and connectedness of D implies that the pair is transitive. It is perhaps surprising that, conversely, if (Ļāā,Ļāā) is a transitive pair then a dessin dāenfant (D,X) can be constructed whose monodromy pair is (Ļāā,Ļāā). Equivalence classes of these transitive pairs are also in bijection with equivalence classes of dessins dāenfants. The equivalence relation for permutation pairs, simultaneous conjugation, is easy to describe and will be immediately important: (Ļāā,Ļāā) is equivalent to (Ļāā²ā,Ļāā²ā) if and only if there is sāSEā such that Ļāā²ā=Ļāsā and Ļāā²ā=Ļāsā. Thus, the theory of Dessins dāEnfants is equivalent to the theory of transitive permutation pairs.
The operation of ānon-simultaneousā conjugation, i.e. the maps (Ļāā,Ļāā)ā¦(Ļāsā,Ļāā) for various sāSEā seems to be important. One reason is that if two dessins dāenfants share certain graph-theoretic data then their monodromy pairs are related by such a conjugation. One of the most elementary facts about the Gal(Qā/Q)-action on dessins dāenfants is that two dessins dāenfants in the same Gal(Qā/Q)-orbit are indeed related in this way. Another equally elementary fact is that the surfaces of two dessins dāenfants in the same Gal(Qā/Q)-orbit have the same genus. So, it seems important ultimately to understand the subset222Any such subset is a union of double cosets: For any CāC(Ļāā)\SEā/C(Ļāā), where C(Ļ) is the centralizer of Ļ, the effect of (Ļāā,Ļāā)ā¦(Ļāsā,Ļāā) is the same for all sāC, since pairs are considered modulo simultaneous conjugation. of sāSEā for which (Ļāsā,Ļāā) is again transitive and to understand which among those preserve genus. Such conjugations can also be seen in the action by Gal(Qā/Q) on the Grothendieck-Teichmüller Group, cf. Proposition 1.6 in [ihara]; it is possible to make precise the connection between these conjugations of the Grothendieck-Teichmüller Group and conjugations of permutations pairs.
Results
For locations of anything mentioned here, see the next subsection Outline.
For (D,X) a dessin dāenfant with edges E and monodromy pair (Ļāā,Ļāā), it is well-known that the connected components of XāD, called faces, are in natural correspondence with ĻāāĻāā-orbits. Since the ā-vertices and ā-vertices correspond to Ļāā-orbits and Ļāā-orbits, the Euler Characteristic ĻXā of X can be computed directly: ĻXā=O(Ļāā)+O(Ļāā)āā£Eā£+O(ĻāāĻāā). It is very useful to generalize this, in the most direct way possible, to all pairs. Let E be a finite set and SEā its symmetric group. For arbitrary Ļāā,ĻāāāSEā, define the synthetic Euler characteristic of (Ļāā,Ļāā) to be
[TABLE]
Inspired by the well-known formula āĻ=2ā2gā, define the synthetic genus of (Ļāā,Ļāā) to be
[TABLE]
There is an operation ā that seems to be well-adapted to the question, which is also a variation on and generalization of the operation of āedge slidingā from Topological Graph Theory, cf. §3.3.3 of [GT].
Definition**.**
For each distinct pair a,bāE, there is a reroute operation ā on SEāĆSEā. The idea of ā is to āunplugā edge a from its ā-vertex and reconnect it to the ā-vertex of b. Any non-simultaneous conjugation (Ļāā,Ļāā)ā¦(Ļāsā,Ļāā) is essentially achieved by repeated application of ā relative to various edges which are easy to read from s.
The following Definition and Theorem, and its proof, are the foundation of the main conclusions.
Definition**.**
Relative to (a,b), call a pair (Ļāā,Ļāā) in SEā
Type U (Unoriented)* iff a,Ļāā(a),b are distinct and no two of them represent the same ĻāāĻāā-orbit.*
-
Type N (Negatively Oriented)* iff a,Ļāā(a),b represent the same ĻāāĻāā-orbit and the cycle containing them is of the form333Here, Ļāā(a)ī =b is required but Ļāā(a)=a is allowed. The right way to talk about this, valuable elsewhere, is via the notion of arc, cf. Definition 3.3. But this omitted from the Introduction.(ā¦aā¦bā¦Ļāā(a)ā¦).*
-
Type P (Positively Oriented)* iff it is neither444āType Pā can be described directly, cf. Definition 5.11, but this is omitted from the Introduction. Type U nor Type N.*
Reroute Theorem**.**
Let (Ļāā,Ļāā) be an arbitrary pair in SEā, not necessarily transitive. Let (Ļāāā,Ļāāā) be the reroute of (Ļāā,Ļāā) relative to (a,b). Let g be the synthetic genus of (Ļāā,Ļāā) and gā that of (Ļāāā,Ļāāā).
(1)
If (Ļāā,Ļāā) is Type U relative to (a,b) then gā=g+1.
2. (2)
If (Ļāā,Ļāā) is Type N relative to (a,b) then gā=gā1.
3. (3)
If (Ļāā,Ļāā) is Type P relative to (a,b) then gā=g.
By studying what happens after repeated application of the reroute operation ā, one can conclude the following answer to the original question:
Definition**.**
Relative to (a,b), call a pair (Ļāā,Ļāā) in SEāExceptional iff a,Ļāā(a),b,Ļāā(b) are distributed into ĻāāĻāā-cycles in one of the following ways:
(ā¦Ļāā(a)ā¦Ļāā(b)ā¦bā¦aā¦)* with Ļāā(a)ī =a*
-
(ā¦Ļāā(b)ā¦Ļāā(a)ā¦aā¦bā¦)* with Ļāā(b)ī =b*
-
(ā¦aā¦bā¦),(ā¦Ļāā(a)ā¦Ļāā(b)ā¦)**
-
(ā¦aā¦Ļāā(a)ā¦),(ā¦bā¦Ļāā(b)ā¦)**
The last of these is called Wild Exceptional, for reasons that are explained later.
Transitivity Theorem**.**
Let (D,X) be a dessin dāenfant with edges E and monodromy pair (Ļāā,Ļāā). Fix distinct a,bāE and let tāSEā be the transposition exchanging a and b.
(1)
If (Ļāā,Ļāā) is not Exceptional relative to (a,b) then (Ļātā,Ļāā) is transitive.
2. (2)
If (Ļāā,Ļāā) is Exceptional then examples show that (Ļātā,Ļāā) may or may not be transitive, depending on the truth of certain connectivity properties of D.
After using the Reroute Theorem to analyze the Exceptional cases, the Transitivity Theorem implies:
Corollary**.**
Let everything be as in the Transitivity Theorem.
If O(ĻātāĻāā)<O(ĻāāĻāā) then (Ļātā,Ļāā) is transitive.
By imposing additional hypotheses, some converses are gained. For example:
Corollary**.**
Let everything be as in the Transitivity Theorem, but assume that X=S.
If (Ļāā,Ļāā) is not Wild Exceptional relative to (a,b) then: (Ļātā,Ļāā) is transitive if and only if O(ĻātāĻāā)ā¤O(ĻāāĻāā).
A strong statement can be given in the admittedly narrow class of trees:
Proposition**.**
Let everything be as in the Transitivity Theorem, but assume D is a tree.
For all sāSEā, (Ļāsā,Ļāā) is transitive if and only if O(ĻāsāĻāā)=1.
Finally, transpositions are classified according to how they change synthetic genus.
Genus Theorem**.**
Let (Ļāā,Ļāā) be an arbitrary pair in SEā, not necessarily transitive. Using descriptions similar in flavor to those of āTypeā or āExceptionalā, transpositions tāSEā are classified according to whether (Ļātā,Ļāā) has higher, lower, or equal synthetic genus than (Ļāā,Ļāā). Regardless, the synthetic genus may change by at most 1.
Overall, the approach is somewhat messy and the reader may reasonably ask if there is a significantly more elegant approach. I think there is not, due to the specificity and opaqueness of (a) the notion of āTypeā, (b) the Exceptional cases, and (c) the classification according to synthetic genus.
In the future, I hope to understand general conjugations as completely as those by transpositions. On the other hand, I think the facts here are sufficient to allow work on a genuinely Galois-theoretic question, restricted to the case of quadratic extensions.
Outline
In §1, I set some notation and recall a few standard facts from the subjects concerned. I also make explicit some conventions that may not be standard.
In §2, I formalize some notions that are likely variations on things that are very well-known. One, a dessin dāfamille, is a natural generalization of a dessin dāenfant which will be very useful (2.1). Additionally, I provide a few basic tools to go with these notions, like their relationship to permutations (2.2, 2.6) and a similarly generalized notion of genus (2.14). As a bonus, a well-known fact about dessins dāenfants whose precise statement and proof does not seem to appear in the literature is generalized and proved (2.10).
In §4, I give a nice description, in terms of the monodromy pair, of those edges of a dessin dāenfant which border only one face instead of two (4.1). For a dessin dāenfant in S, it is equivalent to say that deletion of the edge results in a disconnected graph, but for dessins dāenfants in surfaces of higher genus, disconnection is merely a sufficient condition. I do not understand at this time how to characterize, in terms of the monodromy pair, those edges whose deletion results in a disconnected graph. An analogous question appears in §7.3, and a good answer to it would significantly improve the Transitivity Theorem.
In §3, which is very short, I define the slightly unusual concept of arc (3.3). Given an element Ļ of a group acting on a set, an arc is essentially a half-open interval in a Ļ-orbit, after arranging the orbit as a circuit with Ļ(x) following x for every x in the orbit. Arcs are used many times in the rest of the paper.
In §5, I define the reroute operations ā on SEāĆSEā (5.1). Every choice of distinct a,bāE yields a different operation, and for any sāSEā it is possible to choose such pairs in E so that (Ļāsā,Ļāā) is the same as performing in succession the reroute operations relative to the chosen pairs (6.2). By using the concept of arc, SEāĆSEā can be perfectly partitioned (5.10) so as to predict exactly how ā will change (5.12) the synthetic genus of a pair.
In §6, I use both the statements and the proofs from §5 to study the repeated application of the ā operation. The results are not conceptual, and are presented essentially as a database to be exploited heavily in §7 and §8. The concept of arc is again valuable here, allowing a very annoying amount of seemingly special cases to be unified. The exceptional classes of permutation pairs, those for which the conclusion of the Transitivity Theorem is not certain, are defined here (6.3, 6.4, 6.5, 6.6).
In §7, the Transitivity Theorem is stated and proved (7.2). Examples are given which illustrate the range of behavior that the exceptional pairs may exhibit. Finally, it is proved that the conclusion of the Transitivity Theorem in the exceptional cases is equivalent to a certain connectivity property which is perhaps closer to āpureā Graph Theory than most other things in this paper (7.5, 7.12).
In §8, I give an explicit description of permutation pairs according to how genus will change after conjugating by a transposition (8.1, 8.3). The results are again not conceptual, and are mostly just a consolidation of the database from §6. The concept of arc is valuable here too.
In the Appendix, some MAGMA functions are provided. Due to the complexity in §5, §6, §7 it seemed appropriate to check the conclusions by computer in a reasonably large symmetric group. These functions were used to do this.
Many examples and pictures are provided throughout the paper. In the spirit of Dessins dāEnfants, and taking into account the familiarity that todayās children have with computers, all pictures were drawn by hand using very rudimentary paint software.
Acknowledgements
Most material here was generated from the summer of 2016 to the winter of 2016/2017. However, my introduction to the subject, the decision to pursue this question, and some important early progress that informed the overall direction, occurred while I was a postdoc at University of Wisconsin and while I was a fellow at the Mathematical Sciences Research Institute (DMS-1440140) in Fall 2014.
I thank Nathan Clement and Ted Dewey for many good conversations about the subject while we were at UW. I thank Brian Hwang for the same, at MSRI and at Cornell. I thank Tom Haines, for his interest in the project and for encouraging me to finish it. I thank Jack Graver and Mark Watkins, for their friendliness to me as a new member of Syracuse University, and for good conversations about Graph Theory. Finally, I thank the computing staff at UW, especially Sara Nagreen and John Heim, for their assistance with MAGMA even after my position at UW ended.
1. Notation and Conventions
The cardinality of a set E is denoted ā£Eā£. The sphere is denoted S and the torus is denoted T. In examples/pictures below, S is always presented as the plane, with the reader expected to imagine the point at infinity, and the orientation is always ācounterclockwiseā. In examples/pictures below, T is always oriented by āright-hand-rule from the outsideā. For X a topological space, Ļ0ā(X) denotes the set of connected components of X. For a group Ī acting on a set E, a subset FāE is Ī-stable iff gā xāF for all xāF; when Ī is cyclic and generated by γ, such a subset is called γ-stable instead. For finite sequences (x1ā,x2ā,ā¦,xnā), rotation is the operation (x1ā,x2ā,ā¦,xnā)ā¦(x2ā,ā¦,xnā,x1ā) and reversal is the operation (x1ā,x2ā,ā¦,xnā)ā¦(xnā,ā¦,x2ā,x1ā).
1.1. Permutations
For E a finite set, SEā denotes its symmetric group. For Ļ,sāSEā, the conjugate sā Ļā sā1 is abbreviated to Ļs.
A Ļ-cycle is a cycle in the disjoint cycle decomposition of Ļ. Trivial cycles (fixed points) are always considered to be legitimate cycles, so the reader must be careful about the sense in which a permutation is considered to be āa cycleā. Cycle notation is used in the customary way: the cycle (a,b,c) sends a to b etc. The operation in SEā is āfunctionalā, so applying Ļ1āĻ2ā to eāE results in Ļ1ā(Ļ2ā(e)).
For more details of everything in this subsection, consult the very excellent book [GT].
By abuse of terminology, the term āgraphā will always mean what is more commonly called a āmultigraphā: it is allowed that there are multiple edges incident to the same pair of vertices. A graph is nondegenerate555I am not aware of any standard terminology for this restriction, although it also appears in some key literature, e.g. [HR]. iff every vertex is incident to at least one edge, and degenerate otherwise. For G a graph and e an edge, Gāe denotes the subgraph obtained by deleting the edge e: Gāe has the same vertices as G and all edges of G except e. Note that Gāe may be degenerate even if G was nondegenerate.
For G a graph and x,y vertices, a walk from x to y means the customary thing: a sequence v0ā,e1ā,v1ā,ā¦,enā,vnā with viā vertices, eiā edges such that eiā is incident to viā1ā and viā for all 0<iā¤n, v0ā=x, and vnā=y. A graph is connected iff there is a walk from x to y for all vertices x,y. The set of connected components of a graph G is denoted Ļ0ā(G).
Let ā and ā be formal symbols, fixed throughout the paper. For a graph G with vertices V, a coloring is a function Vā{ā,ā}. If a coloring is fixed then vāV is called a ā-vertex (resp. ā-vertex) iff the image of v under Vā{ā,ā} is ā (resp. ā).
A bicolored graph is a pair (G,f) where G is a graph and f is a coloring such that every edge is incident to both a ā-vertex and a ā-vertex. This differs from the notion of ābipartiteā only in that a choice of color for each vertex is fixed. For e an edge, its ā-vertex will be denoted āeā and its ā-vertex āeā. Note that bicolored graphs have no loops. Throughout the rest of the paper, the coloring function will be suppressed from the notation without exception.
1.3. Dessins dāEnfants
For more details of everything in this subsection, consult the very excellent book [GG].
A dessin dāenfant is a triple (D,X,ι) with X a connected oriented compact surface without boundary, D a finite bicolored nondegenerate666Since D is necessarily connected, this extra condition really only excludes one trivial case: a single vertex in S. Nonetheless, something must be assumed, and nondegeneracy seems the best expression. graph, ι an embedding DāŖX, and it is required that Xāι(D) is homeomorphic to a union of open discs, each of which is called a face. Necessarily, D is a connected graph. Usually the embedding ι will be omitted from the notation. Dessins dāenfants (D1ā,X1ā) and (D2ā,X2ā) are isomorphic iff there is an orientation-preserving homeomorphism X1āāX2ā which induces a graph isomorphism D1āā¶ā¼āD2ā.
Associated with a dessin dāenfant (D,X) is a pair (Ļāā,Ļāā), called the monodromy pair777This pair indeed defines a representation of a fundamental group ā see §4.3.1 of [GG]., in SEā, where E are the edges of D. Necessarily, the pair (Ļāā,Ļāā) is transitive. The function (D,X)ā¦(Ļāā,Ļāā) induces a bijection between isomorphism classes of dessins dāenfants and equivalence classes of transitive pairs in SEāĆSEā modulo the equivalence relation of āsimultaneous conjugationā, i.e. (Ļāā²ā,Ļāā²ā) is equivalent to (Ļāā,Ļāā) iff there is sāSEā such that Ļāā²ā=Ļāsā and Ļāā²ā=Ļāsā. By construction of the function (D,X)ā¦(Ļāā,Ļāā), the ā-vertices of D are in natural bijection with Ļāā-orbits and the ā-vertices with Ļāā-orbits. It is also true, though less obvious, that the faces of (D,X) are in natural bijection with ĻāāĻāā-orbits. If (D,X) is a dessin dāenfant with E the edges of D then it follows that the Euler characteristic Ļ of X can be computed by the following formula: Ļ=O(Ļāā)+O(Ļāā)āā£Eā£+O(ĻāāĻāā). An important feature of the bijection is that if F is a face and O is the corresponding ĻāāĻāā-orbit then the edges bordering F are OāŖĻāā(O). A precise statement and proof of the correspondence between faces and ĻāāĻāā-orbits seems not to appear in print, and a generalization of it will be needed anyway, so a proof is included in §2 of this article, using the content of [HR].
2. Dessins dāFamilles and Genus
Most of this section is, on some essential level, well-known and not new at all. However, some things do not appear in print and other things are not tailored to the goals here. So, §2 is used to set some terminology and record some basic facts.
It will be necessary to work with something more general than a dessin dāenfant:
Definition 2.1** (Dessin dāFamille).**
A dessin dāfamille is a triple (G,X,ī·) where X is a connected oriented compact surface without boundary, G is a bicolored graph, and ī·:GāŖX is an embedding. It is nondegenerate iff G is nondegenerate. Usually the embedding ī· will be omitted from the notation.
Note that it is not assumed that XāG is homeomorphic to a finite union of open discs, nor even that G is connected. Nonetheless, the surface X allows one to extract from G something like the monodromy pair of a dessin dāenfant:
Definition 2.2** (Monodromy).**
For a dessin dāfamille (G,X) with E the edges of G, the monodromy pair of (G,X) is the pair (Ļāā,Ļāā) in SEā where Ļāā is the permutation expressing the cyclic ordering of the edges incident to each ā-vertex according to the orientation of X and Ļāā is the analogous permutation relative to the ā-vertices.
If desired, this can be made rigorous using the Neighborhood Theorem 3.1 in [HR].
An easy but important fact is the equivalence of transitivity and connectedness:
Lemma 2.3** (Transitivity/Connectedness).**
If (G,X) is a dessin dāfamille and G is connected then its monodromy pair (Ļāā,Ļāā) is transitive. If G is nondegenerate then the converse is true.
The idea here is nearly identical to that for dessins dāenfants.
There is a subtle but important difference between a connected dessin dāfamille and a dessin dāenfant. If (G,X,ī·) is a dessin dāfamille with edges E and G is connected then, by Lemma 2.3, its monodromy pair (Ļāā,Ļāā) is transitive. As mentioned in §1, a dessin dāenfant can be constructed from this transitive pair (Ļāā,Ļāā). The subtlety is that the topological surface of this dessin dāenfant can be different from X, although the underlying graph is the same and its embedding has much in common with ī·. I like to say that a connected dessin dāfamille is āa dessin dāenfant in the wrong surfaceā. The following example should make the idea clear:
Example 2.4*.*
Consider the dessin dāfamille (G,T) in the following picture:
This G is connected but (G,T) is not a dessin dāenfant, since the complement TāG has a connected component which is not simply connected. The monodromy pair (Ļāā,Ļāā) of (G,T) has disjoint cycle decompositions Ļāā=(1,2,3)ā (4,5) and Ļāā=(1)ā (2)ā (3,4,5). The pair (Ļāā,Ļāā) is certainly transitive, and if one were to construct the corresponding dessin dāenfant one would get the same graph embedded into S. Because the monodromy pair is the same, the āconfigurationā of the graph would be the same in S as it is in T.
Definition 2.5** (Models).**
For a finite set E and arbitrary pair (Ļāā,Ļāā) in SEā, a dessin dāfamille (G,X) is a model for (Ļāā,Ļāā) iff E is the edge set of G and (Ļāā,Ļāā) is the monodromy pair of (G,X).
It is an easy formality that models exist for all pairs:
Proposition 2.6** (Models Exist).**
If (Ļāā,Ļāā) is an arbitrary pair in SEā then a nondegenerate model (G,X) exists.
In the rest of the paper, this Proposition will be used without explicit reference.
Recall from §1.3 that if (D,X) is a dessin dāenfant then the number of components of XāD is equal to O(ĻāāĻāā). This can be generalized very naturally to dessins dāfamilles, although to do so rigorously requires the machinery of [HR]. However, it will be helpful to state a vague version first:
Theorem** (preliminary version of Theorem 2.10).**
Let (G,X) be a nondegenerate dessin dāfamille with edges E and monodromy pair (Ļāā,Ļāā).
The complement XāG is a disjoint union of connected components which, as open subsets of X, are surfaces without boundary. Intuitively, each of these connected components can be ācompletedā to a surface with boundary, collectively forming a (likely disconnected) surface with boundary XāGā.
Assertion:* The connected components of the manifold boundary āXāGā are in bijection with the set E/ĻāāĻāā of ĻāāĻāā-orbits and surject onto the connected components of the graph G:*
[TABLE]
This is a generalization because if (D,X) is a dessin dāenfant then ā£Ļ0ā(D)ā£=1 and the components of XāD are homeomorphic to discs, so Ļ0ā(āXāDā) is in canonical bijection with Ļ0ā(XāD).
Example 2.7*.*
The following depicts a disconnected dessin dāfamille (G,T):
Its monodromy pair is Ļāā=(1,2,3)ā (4) and Ļāā=Ļāā, so ĻāāĻāā=(1,3,2)ā (4). The complement TāG is homeomorphic to a cylinder (or annulus), and the two cycles of ĻāāĻāā correspond to the two boundary circles of the cylinder.
Despite the intuitive nature of the claim, it is surprisingly difficult to prove. To justify the inclusion of such a proof here, note the following: It seems that even the well-known version for Dessins dāEnfants has never been proved rigorously in print.
I now review [HR], state a precise version of the Theorem, and prove it. Let
[TABLE]
be the ācompletionā of XāG in the sense of Scissors Theorem 2.3 in [HR]. The space XāGā is a compact surface with boundary, likely disconnected, and Īø is a continuous surjection with various properties. Among those properties are the fact that Īø sends āXāGā onto G and restricts to a homeomorphism between the interior of XāGā and XāG. For more details, consult Theorem 2.3 in [HR].
Since manifolds with boundary are involved, it is necessary to talk about both planes and half-planes, and [HR] uses C as plane and denotes by C+ā the closed upper half-plane, so that R=āC+ā. The main ingredient needed to construct XāGā is a certain set Ī of half-plane maps999Some of these Ī» may not be embeddings. This possibility is allowed so that vertices with valence 1 can be treated. For details, consult Step 1 in the proof of Theorem 2.3 in [HR]. Ī»:C+āāX. There is a natural equivalence relation on the disjoint union of XāG with C+āĆĪ, and XāGā is the quotient space. I denote by Ī»[z] the point of XāGā represented by (z,Ī»)āC+āĆĪ, and by Ī»iā[R] the set of Ī»[x] for all xāR. Roughly speaking, the open half-planes Ī»[C+āāR] overlap to form a collar of G in X and the open intervals Ī»[R] attached to that collar overlap to form āXāGā.
I also attach a few keywords to notions from the important Neighborhood Theorem 3.1 in [HR]. For a point xāG, not necessarily a vertex, a standard neighborhood of x is any of the topological embeddings h:CāX guaranteed by Theorem 3.1 of [HR]. The image h(C)āX is necessarily open, h(0)=x, and x is called the center of h. By construction, the preimage hā1(G) is the set of all re2Ļik/n for all Real rā„0 and all kāZ, for n the valence101010If x is not a vertex then the valence is defined by [HR] to be 2. This is done in order to recognize that a small disc around such x is separated by G into two components, half-discs. of x. This subset hā1(G) is called the star of h and, for fixed kāZ, the set of all re2Ļik/n for all r>0 is called a ray of h. By construction, no edge of G is fully contained in h(C), and no vertex of G is contained in h(C) except possibly x, so for each ray r of h there is eāE such that h(r)āe. A connected component of the complement Cāhā1(G) is called a cone of h. Given a fixed orientation of C, the cones of h are cyclically ordered and, for each ray r, there is a cone precedingr and a cone followingr (which are the same cone iff n=1). I will frequently use the following fact, from Step 6 of the proof of Theorem 2.3 in [HR]: If h is a standard neighborhood and C is a cone of h then there exists γ:C+āāC, called suitable below, such that hāγāĪ.
There is a natural surjection, essentially just a restriction of Īø, that will be important:
Definition 2.8**.**
The function
[TABLE]
sends J to the connected component of G containing the (necessarily connected) subset Īø(J)āG. It is immediate from Theorem 3.2(c) in [HR] that āĪø is surjective.
Using the orientation of X, another important natural function can be defined. A bit more work is necessary to define it, and this will precede the formal definition.
Let eāE be arbitrary. Let h:CāX be a standard neighborhood of the vertex āeā. Let rāC be the ray of h for which h(r)āe, and let CāC be the cone following r. As in Step 6 of the proof of Theorem 2.3 in [HR], setting Ī»=defhāγ for suitable γ:C+āāC yields a point Ī»[0]āāXāGā. According to the equivalence relation defining XāGā, the connected component JāāXāGā containing Ī»[0] is independent of h and γ.
Definition 2.9**.**
The function
[TABLE]
sends e to the connected component J described in the previous paragraph.
Note that āĪøārāā:EāĻ0ā(G) is the obvious function.
Remark*.*
Roughly speaking, Ļāā rotates e to Ļāā(e) according to the orientation of X and, in doing so, āsweeps outā a small cone in XāG bounded by e and Ļāā(e). This cone lifts to a small half-plane in XāGā, whose boundary is inside rāā(e).
It will also be convenient to have the counterpart to rāā, the function
[TABLE]
defined by repeating the construction of rāā with ā and ā exchanged.
Theorem 2.10**.**
Denote by E/ĻāāĻāā the set of ĻāāĻāā-orbits.
Assertions:**
(1)
rāā* is surjective.*
2. (2)
rāā* is constant on each ĻāāĻāā-orbit.*
3. (3)
rāā* restricts to a bijection rāā:E/ĻāāĻāāā¶ā¼āĻ0ā(āXāGā).*
4. (4)
For J=rāā(e), the sequence of edges occurring in the combinatorial boundary associated by [HR] to J is, modulo rotation and reversal, e,Ļāā(e),ĻāāĻāā(e),ā¦.
In particular, the set of edges occurring in Īø(J)āG is OāŖĻāā(O) for O the ĻāāĻāā-orbit such that rāā(O)=J.
All assertions are still true after exchanging ā and ā.
Proof.
assertion (1) Let JāĻ0ā(āXāGā) be arbitrary and let xāJ be a point. By Step 7 of the proof of Theorem 2.3 in [HR], there is a standard neighborhood h:CāX and a cone C of h such that precomposing h with suitable C+āāC yields Ī»āĪ such that Ī»[0]=x. Let rāC be the ray of h for which C follows r, and let eāE be such that h(r)āe. It follows from the definition of rāā that rāā(e)=J, so rāā is surjective.
assertion (2) Let eāE be arbitrary, and set ϵ=defĻāā(e). For each point of ϵ, choose a standard neighborhood of that point, and select from this open cover a finite subcover h1ā,h2ā,ā¦,hnā. By choice, hiā(0)ī =hjā(0) for all iī =j. Since standard neighborhoods do not contain any vertices of G except possibly their centers, there must be i,j such that hiā(0)=āϵā and hjā(0)=āϵā. After renumbering if necessary, I can assume that h1ā(0)=āϵā and hnā(0)=āϵā. Fix a parameterization of ϵ by [0,1] such that 0ā¦āϵā and 1ā¦āϵā, and transport to ϵ the usual total order on [0,1]. After renumbering if necessary, I can assume that h1ā(0)<h2ā(0)<āÆ<hnā(0).
I now choose a special cone ciā for each hiā. Let r1āāC be the ray of h1ā such that h1ā(r1ā)āϵ and let c1āāC be the cone of h1ā preceding r1ā. Similarly, let rnāāC be the ray of hnā such that hnā(rnā)āϵ and let cnāāC be the cone of hnā following rnā. Now, suppose 0<i<n. Since hiā is standard and hiā(0)āϵ, the star of hiā is simply R and hiā(R)āϵ. The total order of ϵ therefore orders the two rays of hiā, the positive and negative axes of R, and ciā is defined to be the cone following whichever ray of hiā is āfirstā according to this order. In other words, ciā is the cone following whichever ray r satisfies hiā1ā(0)<hiā(x)<hiā(0) for all xār. For each i, precomposing hiā with a suitable C+āāciā yields Ī»iāāĪ.
I claim that the union of Ī»iā[R]āāXāGā for all i is a connected subset of āXāGā. Since Ī»1ā[0]ārāā(e) by definition of rāā and choice of cone c1ā, and since Ī»nā[0]ārāā(ϵ) by definition of rāā and choice of cone cnā, this shows that rāā(Ļāā(e))=rāā(e). Exchanging colors and applying again shows that rāā(ĻāāĻāā(e))=rāā(Ļāā(e)). Therefore, rāā(ĻāāĻāā(e))=rāā(e), i.e. rāā is constant on each ĻāāĻāā-orbit.
assertion (4) Recall from §5 of [HR] that the combinatorial boundary associated to any JāĻ0ā(āXāGā) is a certain closed walk PJā in G such that Īø(J)=PJā. Fix eāE and set J=defrāā(e). As in [HR], the circle J is partitioned by finitely many points xiā into finitely many arcs siā so that each Īø(xiā) is a vertex of G and each Īø(siā) is an edge of G. Since G is bicolored, there are at least two xiā and at least two siā. It is clear from the definition of Ī»iā above that the image by Īø of the open interval Ī»1ā[R]āŖĪ»2ā[R]āŖāÆāŖĪ»nā[R]āJ intersects the edges e,Ļāā(e),ĻāāĻāā(e). By construction of the combinatorial boundary PJā, the sequence (modulo rotation and reversal) EJā of edges occurring in PJā contains e,Ļāā(e),ĻāāĻāā(e) as a subsequence. Repeating the argument proves the last assertion in the Theorem. It remains to prove that rāā is injective on ĻāāĻāā-orbits.
assertion (3) It follows from Theorem 2.3(d) in [HR] and the construction of the combinatorial boundary in [HR] that each eāE appears twice among the combinatorial boundaries: either once in both EJā and EJā²ā for some distinct J,Jā² or twice in EJā for unique J. In other words, 21āāJāā£EJāā£=ā£Eā£. It follows from (4) that ā£Oā£=21āā£EJāā£, for O the ĻāāĻāā-orbit such that rāā(O)=J. Since rāā is already known to be surjective by (1), failure of injectivity would imply āOāā£Oā£>āJā21āā£EJāā£, a contradiction due to ā£Eā£=āOāā£Oā£.
ā
Corollary 2.11**.**
O(ĻāāĻāā)ā„ā£Ļ0ā(G)ā£.
Proof.
By Theorem 2.10, āĪøārāā is a surjection E/ĻāāĻāāā Ļ0ā(G).
ā
In particular, Ī»1ā[x1ā]=Ī»2ā[x2ā] in XāGā, where Ī»iā is the precomposition of hiā with a suitable C+āāCiā.
Proof.
The last claim follows from the first by definition of the equivalence relation used to construct XāGā, Step 2 of the proof of Theorem 2.3 in [HR].
Let D1āāC be an open disc centered at x1ā, so small that D1ā does not intersect any other ray of h1ā and h1ā(D1ā)āh2ā(C). Since h1ā(C)āX is open, there is an open disc D2āāC centered at x2ā such that h2ā(D2ā)āh1ā(D1ā). Necessarily, Diāāriā consists of two connected components, one of which is contained in the cone Ciā. Call that component Hiā, and note that there is a homeomorphism C+āāHiā which sends [math] to xiā and restricts to RāāHiā. Thus, the goal is to show that h2ā(H2ā)āh1ā(H1ā), since then precomposing h2ā with the homeomorphism yields a closed half disc C+āāŖX with the desired properties.
For any triangle T1āāC with edge I1ā and opposite vertex [math], the orientation of its boundary induced by X via h1ā orders the two connected components of I1āā{x1ā}. By definition of āprecedingā, the component contained in C1ā is that which is considered āfirstā by this ordering. Similarly, any triangle T2āāC with edge I2ā and opposite vertex [math] orders the two connected components of I2āā{x2ā} and, by definition of āfollowingā, the component contained in C2ā is that which is considered āsecondā. Thus, the goal is to show that there are T1ā,T2ā such that h1ā(T1ā) and h2ā(T2ā) induce opposite orientations on I.
Now, let t1āāC be such that h2ā1ā(h1ā(t1ā)) is the triangle in Q complementary to t2ā. So, t1ā is a triangle with edge I1ā and opposite vertex v1āār1ā. I claim that v1ā is between x1ā and [math] on the ray r1ā, which finishes the proof: it is clear that there is a triangle T1ā with edge I1ā and opposite vertex [math] such that t1āāT1ā, therefore T1ā and t1ā induce the same ordering of Ļ0ā(I1āā{x1ā}), thus the goal is to prove that h(t1ā) and h(t2ā) induce opposite orientations of I, which is clear since h(t1ā),h(t2ā)āX are simplices intersecting only along I.
Exactly one of v1ā and h1ā1ā(h2ā(v2ā)) is between [math] and x1ā. Suppose for contradiction that h1ā1ā(h2ā(v2ā)) is, instead of v1ā. Let E1āā[0,1] be the closed segment on r1ā from [math] to x1ā. Let E2āā[0,1] be the closed segment on r2ā from [math] to x2ā. By definition of āstandardā, there is an edge ϵ of G such that h1ā(r1ā),h2ā(r2ā)āϵ. The restrictions hiā:Eiāāϵ are continuous injections [0,1]āŖR, therefore monotone by a corollary of the Intermediate Value Theorem. Since h1ā(x1ā)=h2ā(x2ā), the images intersect. By the contradiction hypothesis, the images share at least two points. An easy argument then shows that either h1ā(E1ā)āh2ā(E2ā) or h1ā(E1ā)āh2ā(E2ā), so either h1ā(0)āh2ā(E2ā) or h2ā(0)āh1ā(E1ā), violating the hypothesis.
ā
A nice fact about trees, hinting at more general statements, can now be proved without too much work:
Proposition 2.13** (Tree Case).**
Let (T,S) be a tree dessin dāenfant with edges E and monodromy pair (Ļāā,Ļāā). Assertion: For arbitrary sāSEā, (Ļāsā,Ļāā) is a transitive pair if and only if O(ĻāsāĻāā)=1.
Proof.
Assume that (Ļāsā,Ļāā) is transitive, so it corresponds to a dessin dāenfant. Let gs be the genus of this dessin dāenfant. In particular, gsā„0. Since the new dessin dāenfant has the same quantity of vertices and the same quantity of edges as T, and since the genus of S is [math], the formula (cf. §1.3) for Euler Characteristic implies that O(ĻāsāĻāā)ā¤O(ĻāāĻāā). Since T is a tree, O(ĻāāĻāā)=1, which forces O(ĻāsāĻāā)=1. For the converse, I prove the contrapositive. Let (G,X) be a nondegenerate model for (Ļāsā,Ļāā). By hypothesis and Lemma 2.3, G is disconnected. By Corollary 2.11, O(ĻāsāĻāā)ā„2.
ā
It will be very useful to have generalizations, for dessins dāfamilles, of Euler characteristic and genus. The following is such, and simply extends without modification the well-known formula from transitive permutation pairs to all permutation pairs:
Definition 2.14** (Synthetic Genus).**
For a finite set E and arbitrary pair (Ļāā,Ļāā) in SEā, the synthetic Euler characteristic of (Ļāā,Ļāā) is
[TABLE]
Accordingly, its synthetic genus is111111Of course, this is inspired by the well-known formula Ļ=2ā2g.
[TABLE]
The synthetic Euler characteristic and synthetic genus of a dessin dāfamille are defined to be those of its monodromy pair.
Clearly, if (G,X) is a dessin dāenfant then its synthetic genus is the genus of X. More generally, if (G,X) is a connected dessin dāfamille then its synthetic genus is the genus of the surface, not necessarily X, into which G embeds as a dessin dāenfant (via its monodromy pair).
Example 2.15*.*
Let (Ļāā,Ļāā) be from Example 2.7. By the information there, the synthetic Euler characteristic is (2+2)ā4+2=2. Thus, this āgenuinely toralā dessin dāfamille has a spherical synthetic Euler characteristic. The important conclusion to draw here is that connectivity and synthetic genus are complementary.
The synthetic Euler characteristic of a dessin dāfamille (G,X,ī·) is the sum of those for (Giā,X,ī·ā£Giāā), where Giā are the connected components of G. The statement can be translated into one about synthetic genus: for example, if (G,X) has synthetic genus g and two connected components (Giā,X) whose synthetic genuses are g1ā and g2ā then g1ā+g2ā=g+1. In particular, synthetic Euler characteristic is always even and synthetic genus is always integral.
3. Sequences and Arcs
Throughout this section, E is a finite set and SEā is its symmetric group.
The following terminology will be convenient in the remainder of the paper:
Definition 3.1** (Sequences).**
For ĻāSEā, a Ļ-sequence is a finite nonempty sequence x0ā,x1ā,ā¦,xnā in E satisfying Ļ(xiā)=xi+1ā for all 0ā¤i<n.
For x,yāE, a Ļ-sequence from x to y is simply a Ļ-sequence x0ā,x1ā,ā¦,xnā such that x0ā=x and xnā=y. It is allowed that x=y even if n>0.
Obviously, if there is a Ļ-sequence from x to y then x,y represent the same Ļ-orbit.
Example 3.2*.*
Define ĻāS4ā by disjoint cycle decomposition: Ļ=def(1,2,3)ā (4). The sequences 1,2,3,1,2 and 4,4,4 are both Ļ-sequences. The minimal Ļ-sequence from 1 to 3 is 1,2,3, while the minimal Ļ-sequence from 1 to 1 is the singleton sequence 1.
The following notion will be used heavily in all remaining sections:
Definition 3.3** (Arcs).**
Let ĻāSEā be arbitrary and x,yāE represent the same Ļ-orbit. It is allowed that x=y. Let x0ā,ā¦,xnā be the unique Ļ-sequence from x to y of minimal length, which is a singleton iff x=y. For xī =y, the Ļ-arc from x to y is defined to be the subsequence x1ā,ā¦,xnā. For x=y, the Ļ-arc from x to y is defined to be the empty sequence.
A key feature of minimal sequences, and therefore also arcs, is that xiāī =x,y for all 0<i<n, although sometimes there are no such i.
Example 3.4*.*
Let Ļ be as in Example 3.2. The Ļ-arc from 1 to 3 is 2,3 and the Ļ-arc from 4 to 4 is the empty sequence.
The notions of āsequenceā and āarcā will be used exclusively for the case that Ļ=ĻāāĻāā for some pair (Ļāā,Ļāā).
It will be useful in §7 to note that if (G,X) is a dessin dāfamille with edges E and monodromy pair (Ļāā,Ļāā) then any ĻāāĻāā-sequence defines a walk in the graph G, as follows.
Suppose x,yāE are in the same ĻāāĻāā-orbit and let x0ā,x1ā,ā¦,xnāāE be a ĻāāĻāā-sequence from x to y. Consider the extended sequence
[TABLE]
Any pair of consecutive edges in sequence (1) shares a well-defined vertex: x0ā and Ļāā(x0ā) are both incident to the same ā-vertex, Ļāā(x0ā) and x1ā=Ļāā(Ļāā(x0ā)) are both incident to the same ā-vertex, etc. Thus, sequence (1) defines a walk from the ā-vertex of x to the ā-vertex of y.
By omitting from (1) the first edge or the last edge or both, one similarly obtains walks from either vertex of x to either vertex of y:
The subsequence Ļāā(x0ā),x1ā,Ļāā(x1ā),ā¦,Ļāā(xnā1ā),xnā defines a walk from the ā-vertex of x to the ā-vertex of y.
-
The subsequence x0ā,Ļāā(x0ā),x1ā,Ļāā(x1ā),ā¦,Ļāā(xnā1ā) defines a walk from the ā-vertex of x to the ā-vertex of y.
-
The subsequence Ļāā(x0ā),x1ā,Ļāā(x1ā),ā¦,Ļāā(xnā1ā) defines a walk from the ā-vertex of x to the ā-vertex of y.
Note also that the edges in (1) are all contained in a single boundary component of XāG. Of course, this is related to āboundary walkā, cf. §3.1.4 of [GT].
Example 3.5*.*
Consider the following dessin dāenfant:
Its monodromy pair (Ļāā,Ļāā) in S5ā has the following disjoint cycle decompositions: Ļāā=(1,2,5,3)ā (4) and Ļāā=(1,2,3)ā (4,5). From this, ĻāāĻāā=(1,5,4,3,2). The minimal ĻāāĻāā-sequence from 1 to 4 is therefore 1,5,4. The extended sequence (1) is 1,2,5,4,4, which yields the following walk: ā,1,ā,2,ā,5,ā,4,ā,4,ā. It is a good idea to visualize this walk in the picture.
4. Incidence and Deletion
Throughout this section, (D,X) represents an arbitrary dessin dāenfant with edges E and monodromy pair (Ļāā,Ļāā).
Recall from §1.3 that faces of (D,X) correspond naturally to ĻāāĻāā-orbits.
Lemma 4.1** (Face Incidence).**
If eāE then the faces of (D,X) bordered by e correspond to those ĻāāĻāā-orbits containing e and Ļāā(e). In particular, e borders only one face of (D,X) if and only if e and Ļāā(e) represent the same ĻāāĻāā-orbit.
Proof.
If F is a face of (D,X) and the corresponding cycle of ĻāāĻāā is c=(x1ā,ā¦,xnā) then, by §1.3, the edges bordering F, with multiplicity, are x1ā,Ļāā(x1ā),ā¦,xnā,Ļāā(xnā). Thus, e borders a face F if and only if its corresponding ĻāāĻāā-cycle c contains an edge x such that either x=e or Ļāā(x)=e. Since a ĻāāĻāā-cycle contains x if and only if it contains ĻāāĻāā(x), and since Ļāā(Ļāā(x))=Ļāā(e), the first statement is proved. The second statement is immediate from the first.
ā
Remark*.*
Although Lemma 4.1 seems to play a very minor role in the rest of the paper, it was actually the observation that led me towards all the other things.
It is well-known that, in the spherical case, deletion of an edge disconnects a graph if and only if the edge borders only one face instead of two. Therefore, I record the following:
Corollary 4.2** (Edge Deletion).**
For eāE, if Dāe is disconnected then e and Ļāā(e) represent the same ĻāāĻāā-orbit. If X=S then the converse is true.
If (D,X) is a dessin dāenfant and Xī =S then it happens frequently that e borders only one face and yet Dāe is connected:
Example 4.3*.*
The following is the trivial dessin dāenfant on T:
This dessin dāenfant has only one face, hence every edge borders only one face, but the graph remains connected after deleting any one of them. Of course, the problem is that circuits in S separate (Jordan Curve Theorem), while circuits in T may not.
It is unclear to me at this time how to characterize these ādisconnectingā edges when X is general in a similarly clean way as Corollary 4.2:
Deletion Question 1*.*
Is there a āgoodā characterization, in terms of the monodromy pair (Ļāā,Ļāā), of those eāE such that Dāe is disconnected?
Such a characterization would be valuable for the classification given in §7 below, especially §7.2 and §7.3.
Nonetheless, it is not too difficult to understand deletion at the level of monodromy if the question of connectivity is ignored. Let (G,X) be a dessin dāfamille with edges E and monodromy pair (Ļāā,Ļāā), and let eāE be arbitrary. Let (Ļāā²ā,Ļāā²ā) be the monodromy pair of the dessin dāfamille (Gāe,X). Disjoint cycle decompositions for Ļāā²ā and Ļāā²ā are obtained from those of Ļāā and Ļāā by deleting e in the obvious way. Because of the important role ĻāāĻāā plays, I give explicit descriptions of Ļāā²āĻāā²ā also, next. Special treatment, which is annoying but not difficult, is needed if Ļāā(e)=e or Ļāā(e)=e, so assume for convenience that Ļāā(e),Ļāā(e)ī =e.
Suppose that e borders only one face. By Lemma 4.1, it is equivalent to suppose that e and Ļāā(e) represent the same ĻāāĻāā-orbit. Let x0ā,x1ā,ā¦,xmāāE be the minimal ĻāāĻāā-sequence from e to Ļāā(e). Since Ļāā(e)ī =e, mā„1. Since also Ļāā(e)ī =e, mā„2. It is easy to verify that c0ā=def(x1ā,ā¦,xmā1ā) is a Ļāā²āĻāā²ā-cycle. Similarly, let y0ā,y1ā,ā¦,ynāāE be the minimal ĻāāĻāā-sequence from Ļāā(e) to e. Since Ļāā(e)ī =e, nā„1. It is easy to verify that c1ā=def(y0ā,ā¦,ynā1ā) is a Ļāā²āĻāā²ā-cycle. Any ĻāāĻāā-cycle that does not contain e is also a Ļāā²āĻāā²ā-cycle and Ļāā²āĻāā²ā is the product of these cycles and c0ā and c1ā. In particular, if Ļ and Ļā² are the synthetic Euler characteristics of (Ļāā,Ļāā) and (Ļāā²ā,Ļāā²ā) then Ļā²=Ļ+2.
Suppose instead that e borders two faces. By Lemma 4.1, it is equivalent to suppose that e and Ļāā(e) represent different ĻāāĻāā-orbits. Let x0ā,ā¦,xmāāE be the ĻāāĻāā-sequence from e to e such that mā1 is the size of the ĻāāĻāā-orbit of e (in particular, mā„2 and {x0ā,ā¦,xmā1ā} is the ĻāāĻāā-orbit of e). Let y0ā,ā¦,ynāāE be the analogous sequence from Ļāā(e) to Ļāā(e). It is easy to verify that c1ā=def(x1ā,ā¦,xmā1ā,y0ā,ā¦,ynā1ā) is a Ļāā²āĻāā²ā-cycle. Any ĻāāĻāā-cycle that contains neither e nor Ļāā(e) is also a Ļāā²āĻāā²ā-cycle and Ļāā²āĻāā²ā is the product of these cycles and c1ā. In particular, Ļā²=Ļ.
Remark*.*
Of course, the statements about Euler characteristics are essentially well-known, e.g. Theorem 3.3.5 in [GT].
It is very important to observe that, even in the āgenericā situation, when Ļāā(e),Ļāā(e)ī =e, two very different situations can result in an increased synthetic Euler Characteristic:
Example 4.4*.*
Let (Ļāā,Ļāā) be as in Example 3.5. The Euler characteristic is, as expected, [math]. Deletion of edge 5 results in Example 2.7, a disconnectedtoral dessin dāfamille with synthetic Euler characteristic 2. On the other hand, deletion of any one of edges 1,2,3 results in a connected dessin dāfamille with Euler characteristic also 2.
5. The Reroute Operation
Throughout this section, E is a finite set, SEā is its symmetric group, distinct a,bāE are fixed, and (Ļāā,Ļāā) is an arbitrary pair in SEā. Despite the notation, (Ļāā,Ļāā) is not assumed to be transitive.
5.1. Definition and goal
Definition 5.1** (Reroute ā).**
Define Eā=def{aāā,aāā}āEā{a}, where aāā and aāā are formal symbols.
Define ĻāāāāSEāā by modifying the disjoint cycle decomposition of Ļāā as follows: Replace a by the symbol aāā and insert the symbol aāā immediately before b in the cycle of Ļāā containing b.
Define ĻāāāāSEāā by modifying the disjoint cycle decomposition of Ļāā as follows: Introduce the trivial cycle fixing aāā and replace a by the symbol aāā.
The pair (Ļāāā,Ļāāā) is called the Reroute of (Ļāā,Ļāā) relative to (a,b).
The relevance of this operation ā to the main question is that to perform a conjugation on (Ļāā,Ļāā) is essentially equivalent to performing a sequence of the operations ā for various choices of a,b. An explicit statement of this for transpositions, which is the only case needed in this paper, occurs as Proposition 6.2 (the general case is not so difficult, but is too notationally cumbersome to justify its inclusion).
Although the most elegant definition of (Ļāāā,Ļāāā) is that given in Definition 5.1 above, it will be convenient to extract some simple facts in the form of a list:
If xī =a and Ļāā(x)ī =a,b then Ļāāā(x)=Ļāā(x).
5. (5)
Ļāāā(aāā)=aāā.
6. (6)
If Ļāā(a)=a then Ļāāā(aāā)=aāā but
otherwise Ļāā(x)=a implies Ļāāā(x)=aāā, and Ļāāā(aāā)=Ļāā(a).
(7)
If xī =a and Ļāā(x)ī =a then Ļāāā(x)=Ļāā(x).
Proof.
This is clear from the definitions of Ļāāā,Ļāāā.
ā
The operation ā is the group-theoretic manifestation of the following picture:
The goal of this section is to determine the exact relationship of O(ĻāāāĻāāā) to O(ĻāāĻāā). This relationship, the Reroute Theorem 5.12 below, and its proof are the technical foundation of the paper.
One reason why this relationship is important is that it predicts genus:
Lemma 5.3** (Genus Change).**
Let (Ļāāā,Ļāāā) be the reroute of (Ļāā,Ļāā) relative to (a,b). Assertion: For Ļ and Ļā the synthetic Euler characteristics of (Ļāā,Ļāā) and (Ļāāā,Ļāāā),
[TABLE]
Equivalently, if g is the synthetic genus of (Ļāā,Ļāā) and gā is that of (Ļāāā,Ļāāā) then
[TABLE]
Proof.
This is obvious: O(Ļāāā)=O(Ļāā), O(Ļāāā)=O(Ļāā)+1, ā£Eāā£=ā£Eā£+1.
ā
If xāE, xī =a and ĻāāĻāā(x)ā/{a,Ļāā(a),b} then xāEā and ĻāāāĻāāā(x)=ĻāāĻāā(x). In particular, if O is an unbiased ĻāāĻāā-orbit then O is also an ĻāāāĻāāā-orbit.
Proof.
It is trivial that xāEā. It is immediate from the hypotheses that Ļāā(x)ī =a, so the definition (7) of Ļāāā implies that Ļāāā(x)=Ļāā(x). The hypotheses further imply that Ļāā(Ļāā(x))ī =a,b, so the definition (4) of Ļāāā implies that Ļāāā(Ļāā(x))=Ļāā(Ļāā(x)). Combining the two equalities yields the first claim. The second claim is immediate from the first.
ā
On the other hand,
Definition 5.6** (Biased).**
An ĻāāāĻāāā-orbit Oā is biased iff it is not also an unbiased ĻāāĻāā-orbit (see Lemma 5.5).
The previous definition is justified by:
Lemma 5.7** (Biased Orbits).**
If Oā is a biased ĻāāāĻāāā-orbit then Oā contains at least one of aāā,aāā,b.
The style of argument here will be repeated many times throughout the rest of the paper. Note that aāā,aāā,b are distinct by definition, but that a,Ļāā(a),b may not be. The possibility that Ļāā(a)ā{a,b} will require special cases to be treated in most of the proofs below. The first example of such a proof is this one.
Proof.
Let Oā be a biased ĻāāāĻāāā-orbit and let xāOā be arbitrary. Note that xī =a because aā/Eā. By nature of the claim, I can also assume that xī =aāā,aāā,b. Finally, I can assume that xī =Ļāā(a): if x=Ļāā(a) then necessarily Ļāā(a)ī =a and so the definitions (1) (5) of Ļāāā,Ļāāā imply that ĻāāāĻāāā(aāā)=Ļāā(a)=x, as desired. In summary, I can assume xāE and xī =a,Ļāā(a),b.
By definition of ābiasedā and Lemma 5.5, the ĻāāĻāā-orbit of x contains at least one of a,Ļāā(a),b. Among all ĻāāĻāā-sequences from x to one of a,Ļāā(a),b, let x0ā,x1ā,ā¦,xnā be the one with minimal length. By the previous paragraph, this sequence is not a singleton (nā„1). By minimality, xiāī =a,Ļāā(a),b for all 0<i<n.
By using Lemma 5.5 repeatedly, x0ā,x1ā,ā¦,xnā1ā is a ĻāāāĻāāā-sequence. To complete the proof, I show that ĻāāāĻāāā(xnā1ā)ā{aāā,aāā,b}. This will be done for each of the three possibilities for xnā.
Suppose first that xnā=Ļāā(a). Since ĻāāĻāā(xnā1ā)=xnā, it follows that Ļāā(xnā1ā)=a. Since xnā1āī =a by the first/second paragraph, definition (6) of Ļāāā says that Ļāāā(xnā1ā)=aāā. Definition (3) of Ļāāā then says that ĻāāāĻāāā(xnā1ā)=b, as desired.
Before treating the other two cases, it will be efficient to make a comment. By the previous paragraph, I can assume that xnāī =Ļāā(a). Since ĻāāĻāā(xnā1ā)=xnā, it follows that Ļāā(xnā1ā)ī =a. Since it is known from the first/second paragraph that xnā1āī =a, definition (7) of Ļāāā says that Ļāāā(xnā1ā)=Ļāā(xnā1ā). Thus, to prove the claim for the remaining two cases it suffices merely to show that Ļāāā(Ļāā(xnā1ā))ā{aāā,aāā,b}.
Suppose now that xnā=b. Since Ļāā(Ļāā(xnā1ā))=xnā and Ļāā(a)ī =xnā=b, definition (2) of Ļāāā says that Ļāāā(Ļāā(xnā1ā))=aāā.
Suppose finally that xnā=a. Since Ļāā(Ļāā(xnā1ā))=xnā and Ļāā(a)ī =xnā=a, definition (1) of Ļāāā says that Ļāāā(Ļāā(xnā1ā))=aāā.
ā
The relevance of all this to the determination of the relationship between O(ĻāāĻāā) and O(ĻāāāĻāāā) is clear:
Proposition 5.8** (Orbit Counting).**
Let U be the number of unbiased ĻāāĻāā-orbits, let B be the number of ĻāāĻāā-orbits containing at least one of a,Ļāā(a),b, and let Bā be the number of ĻāāāĻāāā-orbits containing at least one of aāā,aāā,b. Assertion:O(ĻāāĻāā)=U+B and O(ĻāāāĻāāā)=U+Bā.
Proof.
This is immediate from Definitions 5.4/5.6 and Lemmas 5.5/5.7.
ā
Thus, the problem is to calculate the difference BāāB. The following can be used to calculate the difference BāāB, and more:
Orbit Transfer Lemma 5.9**.**
Let (Ļāāā,Ļāāā) be the reroute of (Ļāā,Ļāā) relative to (a,b).
A ĻāāĻāā-sequence x0ā,ā¦,xnā is called strict relative to (a,b) iff it contains at least two terms (nā„1) and xiāī =a,Ļāā(a),b for all 0<i<n. It is allowed that x0ā=xnā.
Assertions:**
(1)
If x0ā,ā¦xiāā¦,xnāāE is a strict ĻāāĻāā-sequence from a to Ļāā(a) then aāā,ā¦xiāā¦,b is a ĻāāāĻāāā-sequence. In particular, aāā and b represent the same ĻāāāĻāāā-orbit. It is allowed that the ĻāāĻāā-sequence has no interior terms: if a,Ļāā(a) is a ĻāāĻāā-sequence then aāā,b is a ĻāāāĻāāā-sequence.
2. (2)
Assume that Ļāā(a)ī =a,b. If x0ā,ā¦xiāā¦,xnāāE is a strict ĻāāĻāā-sequence from Ļāā(a) to b then aāā,Ļāā(a),ā¦xiāā¦,aāā is a ĻāāāĻāāā-sequence. In particular, aāā and aāā represent the same ĻāāāĻāāā-orbit. It is allowed that the ĻāāĻāā-sequence has no interior terms: if Ļāā(a),b is a ĻāāĻāā-sequence then aāā,Ļāā(a),aāā is a ĻāāāĻāāā-sequence. If Ļāā(a)=b then ĻāāāĻāāā(aāā)=aāā. If Ļāā(a)=a then the conclusion is definitely false.
3. (3)
Assume that Ļāā(a)ī =a. If x0ā,ā¦xiāā¦,xnāāE is a strict ĻāāĻāā-sequence from b to a then b,ā¦xiāā¦,aāā is a ĻāāāĻāāā-sequence. In particular, b and aāā represent the same ĻāāāĻāāā-orbit. It is allowed that the ĻāāĻāā-sequence has no interior terms: if b,a is a ĻāāĻāā-sequence then b,aāā is a ĻāāāĻāāā-sequence.If Ļāā(a)=a then the conclusion is definitely false.
4. (4)
If x0ā,ā¦xiāā¦,xnāāE is a strict ĻāāĻāā-sequence from b to Ļāā(a) then b,ā¦xiāā¦,b is a ĻāāāĻāāā-sequence. In particular, b represents a different ĻāāāĻāāā-orbit than both aāā and aāā. It is allowed that the ĻāāĻāā-sequence has no interior terms: if b,Ļāā(a) is a ĻāāĻāā-sequence then b,b is a ĻāāāĻāāā-sequence, i.e. b is fixed by ĻāāāĻāāā.
5. (5)
Assume Ļāā(a)ī =a,b. If x0ā,ā¦xiāā¦,xnāāE is a strict ĻāāĻāā-sequence from Ļāā(a) to a then aāā,Ļāā(a),ā¦xiāā¦,aāā is a ĻāāāĻāāā-sequence. In particular, aāā represents a different ĻāāāĻāāā-orbit than both aāā and b. It is allowed that the ĻāāĻāā-sequence has no interior terms: if Ļāā(a),a is a ĻāāĻāā-sequence then aāā,Ļāā(a),aāā is a ĻāāāĻāāā-sequence. If Ļāā(a)=a then ĻāāāĻāāā(aāā)=(aāā). If Ļāā(a)=b then the conclusion is definitely false.
6. (6)
Assume that Ļāā(a)ī =b. If x0ā,ā¦xiāā¦,xnāāE is a strict ĻāāĻāā-sequence from a to b then aāā,ā¦xiāā¦,aāā is a ĻāāāĻāāā-sequence. In particular, aāā represents a different ĻāāāĻāāā-orbit than both aāā and b. It is allowed that the ĻāāĻāā-sequence has no interior terms: if a,b is a ĻāāĻāā-sequence then aāā,aāā is a ĻāāāĻāāā-sequence, i.e. aāā is fixed by ĻāāāĻāāā. If Ļāā(a)=b then the conclusion is false.
7. (7)
If x0ā,ā¦xiāā¦,xnāāE is a strict ĻāāĻāā-sequence from a to a then aāā,ā¦xiāā¦,aāā is a ĻāāāĻāāā-sequence. In particular, aāā and aāā represent the same ĻāāāĻāāā-orbit. It is allowed that the ĻāāĻāā-sequence has no interior terms: if a,a is a ĻāāĻāā-sequence then aāā,aāā is a ĻāāāĻāāā-sequence.
8. (8)
If x0ā,ā¦xiāā¦,xnāāE is a strict ĻāāĻāā-sequence from Ļāā(a) to Ļāā(a) then aāā,Ļāā(a),ā¦xiāā¦,b is a ĻāāāĻāāā-sequence. In particular, aāā and b represent the same ĻāāāĻāāā-orbit. It is allowed that the ĻāāĻāā-sequence has no interior terms: if Ļāā(a),Ļāā(a) is a ĻāāĻāā-sequence then aāā,Ļāā(a),b is a ĻāāāĻāāā-sequence.
9. (9)
If x0ā,ā¦xiāā¦,xnāāE is a strict ĻāāĻāā-sequence from b to b then b,ā¦xiāā¦,aāā is a ĻāāāĻāāā-sequence. In particular, b and aāā represent the same ĻāāāĻāāā-orbit. It is allowed that the ĻāāĻāā-sequence has no interior terms: if b,b is a ĻāāĻāā-sequence then b,aāā is a ĻāāāĻāāā-sequence.
Proof.
assertion (1) If the ĻāāĻāā-sequence has only two terms then necessarily Ļāā(a)=a and so it is immediate from the definitions (3) (6) of Ļāāā,Ļāāā that ĻāāāĻāāā(aāā)=b. So, I can assume from now on that nā„2. It follows that Ļāā(x0ā)ī =a and Ļāā(Ļāā(x0ā))ī =a,b since otherwise x1āā{a,Ļāā(a),b} and the assumption āstrictā would be contradicted. Thus, Ļāāā(aāā)=Ļāā(x0ā) by the definition (6) of Ļāāā, and Ļāāā(Ļāā(x0ā))=Ļāā(Ļāā(x0ā)) by the definition (4) of Ļāāā. Combined, aāā,x1ā is a ĻāāāĻāāā-sequence. Using āstrictā again and Lemma 5.5 repeatedly, aāā,x1ā,ā¦,xnā1ā is a ĻāāāĻāāā-sequence. It remains to show that ĻāāāĻāāā(xnā1ā)=b. Since ĻāāĻāā(xnā1ā)=xnā=Ļāā(a), we have Ļāā(xnā1ā)=a. Since xnā1āī =a by āstrictā and nā„2, the definition (6) of Ļāāā implies that Ļāāā(xnā1ā)=aāā. The definition (3) of Ļāāā then implies that ĻāāāĻāāā(xnā1ā)=b.
assertion (2) Because of the assumption Ļāā(a)ī =a, the definitions (1) (5) of Ļāāā,Ļāāā imply that ĻāāāĻāāā(aāā)=Ļāā(a). Thus, it suffices to show that Ļāā(a),ā¦xiāā¦,aāā is a ĻāāāĻāāā-sequence. Note that Ļāā(Ļāā(a))ī =a because otherwise Ļāā(a) would be a fixed point of ĻāāĻāā, contradicting the fact that Ļāā(a) represents the same ĻāāĻāā-orbit as bī =Ļāā(a). Together with the assumption x0ā=Ļāā(a)ī =a, the definition (7) of Ļāāā says that Ļāāā(x0ā)=Ļāā(x0ā). Suppose that the ĻāāĻāā-sequence contains only two terms, i.e. ĻāāĻāā(x0ā)=b. Using the assumption Ļāā(a)ī =b, the definition (2) of Ļāāā says that Ļāāā(Ļāā(x0ā))=aāā. Combining with the known Ļāāā(x0ā)=Ļāā(x0ā) shows that Ļāā(a),aāā is a ĻāāāĻāāā-sequence. Suppose now that the ĻāāĻāā-sequence contains at least three terms (nā„2). Since x0ā=Ļāā(a)ī =a by assumption and xiāī =a,Ļāā(a),b for all 0<i<n by āstrictā, Lemma 5.5 says that x0ā,x1ā,ā¦,xnā1ā is a ĻāāāĻāāā-sequence. It remains to show that ĻāāāĻāāā(xnā1ā)=aāā. It must be true that Ļāā(xnā1ā)ī =a, because otherwise b=xnā=ĻāāĻāā(xnā1ā)=Ļāā(a), contradicting the assumption Ļāā(a)ī =b. Since xnā1āī =a by āstrictā and nā„2, the definition (7) of Ļāāā implies that Ļāāā(xnā1ā)=Ļāā(xnā1ā). Using the assumption Ļāā(a)ī =b and the fact that Ļāā(Ļāā(xnā1ā))=b, the definition (2) of Ļāāā implies that Ļāāā(Ļāā(xnā1ā))=aāā. Combining the two equalities yields ĻāāāĻāāā(xnā1ā)=aāā. The last claim is easy: if Ļāā(a)=b then it is immediate from the definitions (2) (5) of Ļāāā,Ļāāā that ĻāāāĻāāā(aāā)=aāā. If Ļāā(a)=a then it is immediate from the definitions (1) (5) of Ļāāā,Ļāāā that ĻāāāĻāāā(aāā)=aāā, so the conclusion cannot possibly be true.
assertion (3) It must be true that Ļāā(b)ī =a, because otherwise ĻāāĻāā(b)=Ļāā(a), contradicting either Ļāā(a)ī =a (if n=1) or the āstrictā assumption (if nā„2). It follows from this and the definition (7) of Ļāāā that Ļāāā(b)=Ļāā(b). Suppose that the ĻāāĻāā-sequence contains only two terms, i.e. that ĻāāĻāā(b)=a. Since Ļāā(a)ī =a is assumed, the definition (1) of Ļāāā implies that Ļāāā(Ļāā(b))=aāā. Combining with the known Ļāāā(b)=Ļāā(b) yields ĻāāāĻāāā(b)=aāā. Suppose now that the ĻāāĻāā-sequence contains at least three terms (nā„2). Since x0ā=bī =a and xiāī =a,Ļāā(a),b by āstrictā, Lemma 5.5 says that x0ā,x1ā,ā¦,xnā1ā is a ĻāāāĻāāā-sequence. It remains to show that ĻāāāĻāāā(xnā1ā)=aāā. By āstrictā and nā„2, xnā1āī =a. Also, Ļāā(xnā1ā)ī =a since otherwise a=xnā=ĻāāĻāā(xnā1ā)=Ļāā(a), contradicting the assumption. These two facts and the definition (7) of Ļāāā imply that Ļāāā(xnā1ā)=Ļāā(xnā1ā). Since Ļāā(a)ī =a by assumption, the fact that ĻāāĻāā(xnā1ā)=xnā=a and the definition (1) of Ļāāā imply that Ļāāā(Ļāā(xnā1ā))=aāā. Combining these two equalities yields ĻāāāĻāāā(xnā1ā)=aāā. If Ļāā(a)=a then it is immediate from the definitions (1) (5) of Ļāāā,Ļāāā that ĻāāāĻāāā(aāā)=aāā, so the conclusion cannot possibly be true.
assertion (4) If the ĻāāĻāā-sequence contains only two terms then necessarily Ļāā(b)=a and so it is immediate from the definitions (3) (6) of Ļāāā,Ļāāā that ĻāāāĻāāā(b)=b. So, I can assume that nā„2. Necessarily Ļāā(b)ī =a since otherwise x1ā=Ļāā(a), contradicting āstrictā and nā„2. Since xiāī =a,Ļāā(a),b for all 0<i<n by āstrictā, Lemma 5.5 says that b,x1ā,ā¦,xnā1ā is a ĻāāāĻāāā-sequence. It remains to show ĻāāāĻāāā(xnā1ā)=b. Since xnā1āī =a by āstrictā and nā„2, and since ĻāāĻāā(xnā1ā)=xnā=Ļāā(a) implies Ļāā(xnā1ā)=a, the definition (6) of Ļāāā says that Ļāāā(xnā1ā)=aāā. Definition (3) of Ļāāā says that ĻāāāĻāāā(xnā1ā)=b.
assertion (5) Since Ļāā(a)ī =a by assumption, the definitions (1) (5) of Ļāāā,Ļāāā say that ĻāāāĻāāā(aāā)=Ļāā(a). So, it suffices to show that Ļāā(a),ā¦xiāā¦aāā is a ĻāāāĻāāā-sequence. It must be true that Ļāā(x0ā)ī =a since otherwise x0ā=Ļāā(a) would be a fixed point of ĻāāĻāā, contradicting the assumption that a represents the same ĻāāĻāā-orbit as Ļāā(a)ī =a. Combined with the assumption x0ā=Ļāā(a)ī =a, the definition (7) of Ļāāā implies that Ļāāā(x0ā)=Ļāā(x0ā). Suppose that the ĻāāĻāā-sequence contains only two terms, i.e. that ĻāāĻāā(Ļāā(a))=a. Since Ļāā(a)ī =a by assumption, the definition (1) of Ļāāā says that Ļāāā(Ļāā(x0ā))=aāā. Since Ļāāā(x0ā)=Ļāā(x0ā) is known, ĻāāāĻāāā(x0ā)=aāā. Suppose now that the ĻāāĻāā-sequence contains at least three terms (nā„2). Since x0ā=Ļāā(a)ī =a by assumption, and xiāī =a,Ļāā(a),b for all 0<i<n by āstrictā, Lemma 5.5 says that Ļāā(a),x1ā,ā¦,xnā1ā is a ĻāāāĻāāā-sequence. It remains to show that ĻāāāĻāāā(xnā1ā)=aāā. It must be true that Ļāā(xnā1ā)ī =a, since otherwise a=xnā=ĻāāĻāā(xnā1ā)=Ļāā(a), contradicting the assumption Ļāā(a)ī =a. Since xnā1āī =a by āstrictā and nā„2, the definition (7) of Ļāāā says that Ļāāā(xnā1ā)=Ļāā(xnā1ā). Since Ļāā(a)ī =a by assumption, the fact that ĻāāĻāā(xnā1ā)=a and the definition (1) of Ļāāā imply that Ļāāā(Ļāā(xnā1ā))=aāā. Combining the two equalities yields ĻāāāĻāāā(xnā1ā)=aāā. The last claim is easy: if Ļāā(a)=a then it is immediate from the definitions (1) (5) of Ļāāā,Ļāāā that ĻāāāĻāāā(aāā)=aāā. If Ļāā(a)=b then it is immediate from the definitions (2) (5) of Ļāāā,Ļāāā that ĻāāāĻāāā(aāā)=aāā, so the conclusion cannot possibly be true.
assertion (6) It must be true that Ļāā(a)ī =a, since otherwise ĻāāĻāā(a)=Ļāā(a), contradicting either the assumption Ļāā(a)ī =b (if n=1) or āstrictā (if nā„2). The definition (6) of Ļāāā then implies that Ļāāā(aāā)=Ļāā(x0ā). Suppose that the ĻāāĻāā-sequence contains only two terms, i.e. that ĻāāĻāā(x0ā)=b. Since Ļāā(a)ī =b by assumption, the fact that ĻāāĻāā(x0ā)=b and the definition (2) of Ļāāā imply that Ļāāā(Ļāā(x0ā))=aāā. Combined with the known Ļāāā(aāā)=Ļāā(x0ā) yields ĻāāāĻāāā(aāā)=aāā. Suppose now that the ĻāāĻāā-sequence contains at least three terms (nā„2). It is known already that Ļāā(a)ī =a, and Ļāā(Ļāā(x0ā))=x1āī =a,b by āstrictā, so definition (4) of Ļāāā implies that Ļāāā(Ļāā(x0ā))=x1ā. Combined with the known Ļāāā(aāā)=Ļāā(x0ā) yields ĻāāāĻāāā(aāā)=x1ā. By āstrictā, Lemma 5.5 says that aāā,x1ā,ā¦,xnā1ā is a ĻāāāĻāāā-sequence. It remains to show that ĻāāāĻāāā(xnā1ā)=aāā. It must be true that Ļāā(xnā1ā)ī =a, since otherwise implies b=xnā=ĻāāĻāā(xnā1ā)=Ļāā(a), contradicting the assumption Ļāā(a)ī =b. Since also xnā1āī =a is known by āstrictā and nā„2, the definition (7) of Ļāāā says Ļāāā(xnā1ā)=Ļāā(xnā1ā). Since Ļāā(a)ī =b by assumption, the fact that ĻāāĻāā(xnā1ā)=b and the definition (2) of Ļāāā imply that Ļāāā(Ļāā(xnā1ā))=aāā. Combining the two equalities yields ĻāāāĻāāā(xnā1ā)=aāā. If Ļāā(a)=b then it is immediate from the definitions (2) (5) of Ļāāā,Ļāāā that ĻāāāĻāāā(aāā)=aāā, so the conclusion cannot possibly be true.
assertion (7) Note that the assumption implies Ļāā(a)ī =a. Further, Ļāā(a)ī =a because otherwise ĻāāĻāā(a)=Ļāā(a), contradicting the assumption. Since Ļāā(a)ī =a, the definition (6) of Ļāāā implies that Ļāāā(aāā)=Ļāā(x0ā). Suppose that the ĻāāĻāā-sequence contains only two terms, i.e. that ĻāāĻāā(a)=a. Since Ļāā(a)ī =a by assumption, the definition (1) of Ļāāā implies that Ļāāā(Ļāā(a))=aāā. Combined with the known Ļāāā(aāā)=Ļāā(x0ā) yields ĻāāāĻāāā(aāā)=aāā. Suppose now that the ĻāāĻāā-sequence contains at least three terms (nā„2). Since Ļāā(x0ā)ī =a is known and Ļāā(Ļāā(x0ā))=x1āī =a,b by āstrictā, the definition (4) of Ļāāā says that Ļāāā(Ļāā(x0ā))=x1ā. Combined with the known Ļāāā(aāā)=Ļāā(x0ā) yields ĻāāāĻāāā(aāā)=x1ā. Since xiāī =a,Ļāā(a),b for all 0<i<n by āstrictā, Lemma 5.5 says that aāā,x1ā,ā¦,xnā1ā is a ĻāāāĻāāā-sequence. It remains to show that ĻāāāĻāāā(xnā1ā)=aāā. It must be true that Ļāā(xnā1ā)ī =a, since otherwise a=xnā=ĻāāĻāā(xnā1ā)=Ļāā(a), contrary to the assumption. Since xnā1āī =a by āstrictā and nā„2, the definition (7) of Ļāāā implies that Ļāāā(xnā1ā)=Ļāā(xnā1ā). Since Ļāā(a)ī =a by assumption, the fact that ĻāāĻāā(xnā1ā)=a and the definition (1) of Ļāāā imply that Ļāāā(Ļāā(xnā1ā))=aāā. Combined, the two equalities yield ĻāāāĻāāā(xnā1ā)=aāā.
assertion (8) By assumption, Ļāā(a)ī =a,b. In particular, the definitions (1) (5) of Ļāāā,Ļāāā imply that ĻāāāĻāāā(aāā)=Ļāā(a). Therefore, it suffices merely to show that Ļāā(a),ā¦xiāā¦,b is a ĻāāāĻāāā-sequence. Suppose that the ĻāāĻāā-sequence contains only two terms, i.e. that ĻāāĻāā(Ļāā(a))=Ļāā(a). Then Ļāā(Ļāā(a))=a and since Ļāā(a)ī =a is known the definition (6) says Ļāāā(Ļāā(a))=aāā. Definition (3) of Ļāāā implies that ĻāāāĻāāā(Ļāā(a))=b. Suppose now that the ĻāāĻāā-sequence contains at least three terms (nā„2). Since x0ā=Ļāā(a)ī =a is known, and since xiāī =a,Ļāā(a),b for all 0<i<n by āstrictā, Lemma 5.5 says that x0ā,x1ā,ā¦,xnā1ā is a ĻāāāĻāāā-sequence. It remains to show that ĻāāāĻāāā(xnā1ā)=b. Since Ļāā(xnā1ā)=a, because Ļāā(a)=xnā=ĻāāĻāā(xnā1ā), and since xnā1āī =a by āstrictā and nā„2, definition (6) of Ļāāā says Ļāāā(xnā1ā)=aāā. Definition (3) of Ļāāā then says ĻāāāĻāāā(xnā1ā)=b.
assertion (9) Note that the assumption implies Ļāā(a)ī =b. Further, Ļāā(b)ī =a, since otherwise ĻāāĻāā(b)=Ļāā(a), contradicting the assumption. Thus, x0ā,Ļāā(x0ā)ī =a and so the definition (7) of Ļāāā implies that Ļāāā(x0ā)=Ļāā(x0ā). Suppose that the ĻāāĻāā-sequence contains only two terms, i.e. that ĻāāĻāā(b)=b. Since Ļāā(a)ī =b is known, the fact that ĻāāĻāā(b)=b and the definition (2) of Ļāāā imply that Ļāāā(Ļāā(b))=aāā. Combining with the known Ļāāā(x0ā)=Ļāā(x0ā) yields ĻāāāĻāāā(b)=aāā. Suppose now that the arc contains at least three terms (nā„2). Since xiāī =a,Ļāā(a),b for all 0<i<n by āstrictā, Lemma 5.5 says that b,x1ā,ā¦,xnā1ā is a ĻāāāĻāāā-sequence. It remains to show that ĻāāāĻāāā(xnā1ā)=aāā. It must be true that Ļāā(xnā1ā)ī =a since otherwise b=xnā=ĻāāĻāā(xnā1ā)=Ļāā(a), contradicting the assumption. Since xnā1āī =a by āstrictā and nā„2, the definition (7) of Ļāāā says that Ļāāā(xnā1ā)=Ļāā(xnā1ā). Since Ļāā(a)ī =b is known, the fact that ĻāāĻāā(xnā1ā)=b and the definition (2) of Ļāāā imply that Ļāāā(Ļāā(xnā1ā))=aāā. Combining the two yields ĻāāāĻāāā(xnā1ā)=aāā.
ā
Remark*.*
The reader may detect some redundancy among the many statements in Orbit Transfer Lemma 5.9. However, they will eventually all be needed in full detail.
5.3. Types and Theorem
The most important classification in the paper is the following:
Definition 5.10** (Type).**
Recall, from §3, the notion of āarcā. The pair (Ļāā,Ļāā) is
Type U (Unoriented) relative to (a,b) iff a,Ļāā(a),b are distinct and no two of them represent the same ĻāāĻāā-orbit.
-
Type N (Negatively Oriented) relative to (a,b) iff a,Ļāā(a),b represent the same ĻāāĻāā-orbit and the ĻāāĻāā-arc from a to b does not contain Ļāā(a).
-
Type P (Positively Oriented) relative to (a,b) iff it is neither Type U nor Type N.
It is easy to see that this is a partition of SEāĆSEā. Before stating and proving the Reroute Theorem 5.12, I subdivide Type P:
Definition 5.11** (P-Subtypes).**
(Ļāā,Ļāā) is
Type P1 relative to (a,b) iff a,Ļāā(a),b represent the same ĻāāĻāā-orbit and the ĻāāĻāā-arc from a to b contains Ļāā(a).
-
Type P2 relative to (a,b) iff a,Ļāā(a) represent the same ĻāāĻāā-orbit, different from that of b.
-
Type P3 relative to (a,b) iff a,b represent the same ĻāāĻāā-orbit, different from that of Ļāā(a).
-
Type P4 relative to (a,b) iff Ļāā(a),b represent the same ĻāāĻāā-orbit, different from that of a.
It is easy to see that this is a partition of Type P. The definition of āTypeā is mysterious but thoroughly justified by:
Reroute Theorem 5.12**.**
Let (Ļāāā,Ļāāā) be the reroute of (Ļāā,Ļāā) relative to (a,b). Let g be the synthetic genus of (Ļāā,Ļāā) and gā that of (Ļāāā,Ļāāā). Assertions:
(1)
If (Ļāā,Ļāā) is Type U relative to (a,b) then gā=g+1.
2. (2)
If (Ļāā,Ļāā) is Type N relative to (a,b) then gā=gā1.
3. (3)
If (Ļāā,Ļāā) is Type P relative to (a,b) then gā=g.
The reader will recall that if a permutation pair is not transitive then its synthetic genus does not quite have the āexpectedā topological meaning; see Example 2.15.
Proof.
By Lemma 5.3, it suffices to show that O(ĻāāāĻāāā)āO(ĻāāĻāā) is equal to ā2 if (Ļāā,Ļāā) is Type U, is equal to 2 if (Ļāā,Ļāā) is Type N, and is equal to [math] if (Ļāā,Ļāā) is Type P. By Proposition 5.8, the desired difference is BāāB where B is the number of ĻāāĻāā-orbits represented by a,Ļāā(a),b and Bā is the number of ĻāāāĻāāā-orbits represented by aāā,aāā,b.
Suppose (Ļāā,Ļāā) is Type U. It is immediate from the definition of āType Uā that B=3. Any two of Orbit Transfer Lemma 5.9 (7) (8) (9) imply Bā=1.
Suppose (Ļāā,Ļāā) is Type N. It is immediate from the definition of āType Nā that B=1. The definition of āType Nā also supplies the hypotheses of Orbit Transfer Lemma 5.9 (4) (6), which then imply Bā=3.
Suppose (Ļāā,Ļāā) is Type P1. It is immediate from the definition of āType P1ā that B=1. The definition of āType P1ā also supplies the hypotheses of Orbit Transfer Lemma 5.9 (1) (3), which then imply Bā=1.
Finally, suppose that (Ļāā,Ļāā) is Type P2 or Type P3 or Type P4. In all three cases, B=2.
If (Ļāā,Ļāā) is Type P2 then Orbit Transfer Lemma 5.9 (9) implies that aāā and b represent the same ĻāāāĻāāā-orbit, while Orbit Transfer Lemma 5.9 (5) implies that aāā represents a different ĻāāāĻāāā-orbit.
If (Ļāā,Ļāā) is Type P3 then Orbit Transfer Lemma 5.9 (8) implies that aāā and b represent the same ĻāāāĻāāā-orbit, while Orbit Transfer Lemma 5.9 (6) implies that aāā represents a different ĻāāāĻāāā-orbit.
If (Ļāā,Ļāā) is Type P4 then Orbit Transfer Lemma 5.9 (7) implies that aāā and aāā represent the same ĻāāāĻāāā-orbit, while Orbit Transfer Lemma 5.9 (4) implies that b represents a different ĻāāāĻāāā-orbit. In all three cases, Bā=2.
ā
For fun, here are some examples:
Example 5.13* (āThetaā).*
In the picture here, a dessin dāenfant in S is shown on the left, with edges labeled a and b.
Observe that the monodromy pair of this dessin dāenfant is Type U relative to (a,b). By applying the reroute ā relative to (a,b), one obtains a new monodromy pair, and a model for it is shown on the right. As predicted by the Reroute Theorem 5.12, the synthetic Euler characteristic of the model is (1+2)ā4+1=0, reflecting the fact that the ātrueā surface of the new dessin dāfamille is T.
Example 5.14* (Tree #1).*
In the picture here, a tree dessin dāenfant in S is shown on the left, with edges labeled a and b.
Observe that the monodromy pair of this dessin dāenfant is Type N relative to (a,b). By applying the reroute ā relative to (a,b), one obtains a new monodromy pair, and a model for it is shown on the right. As predicted by the Reroute Theorem 5.12, the synthetic Euler characteristic of the model is (2+4)ā5+3=4, higher by 2 than the original Euler characteristic (2+3)ā4+1=2.
Example 5.15* (Tree #2).*
In the picture here, the same tree is used as in the previous Example 5.14, but now a different edge a is chosen.
Observe that the monodromy pair of this dessin dāenfant is Type P1 (since tree dessins dāenfants have only one face, the monodromy pair is always either Type N or Type P1 relative to any pair of edges) relative to (a,b). By applying the reroute ā relative to (a,b), one obtains a new monodromy pair, and a model for it is shown on the right. As predicted by the Reroute Theorem 5.12, the synthetic Euler characteristic of the model is (2+4)ā5+1=2, the same as the Euler characteristic of the original.
Remark*.*
One can see the basic idea of Proposition 2.13 in Examples 5.14 and 5.15: disconnection cannot occur without creating additional circuits.
Example 5.16*.*
In the picture here, a dessin dāenfant in S is shown on the left, with edges labeled a and b.
Observe that the monodromy pair of this dessin dāenfant is Type P2 relative to (a,b). By applying the reroute ā relative to (a,b), one obtains a new monodromy pair, and a model for it is shown on the right. As predicted by the Reroute Theorem 5.12, the synthetic Euler characteristic of the model is (2+2)ā4+2=2, the same as the original Euler characteristic (2+1)ā3+2=2.
Example 5.17* (Circuit #1).*
In the picture here, a circuit dessin dāenfant in S is shown on the left, with edges labeled a and b.
Observe that the monodromy pair of this dessin dāenfant is Type P3 relative to (a,b). By applying the reroute ā relative to (a,b), one obtains a new monodromy pair, and a model for it is shown on the right. As predicted by the Reroute Theorem 5.12, the synthetic Euler characteristic of the model is (2+3)ā5+2=2, the same as the Euler characteristic of the original.
Example 5.18* (Circuit #2).*
In the picture here, the same circuit as in the previous Example 5.17 is used, but now a different edge b is chosen.
Observe that the monodromy pair of this dessin dāenfant is Type P4 relative to (a,b). By applying the reroute ā relative to (a,b), one obtains a new monodromy pair, and a model for it is shown on the right. As predicted by the Reroute Theorem 5.12, the synthetic Euler characteristic of the model is (2+3)ā5+2=2, the same as the Euler characteristic of the original.
5.4. More about models
As usual, let (Ļāāā,Ļāāā) be the reroute of (Ļāā,Ļāā) relative to (a,b).
It will be helpful in §7 to know a bit about models for (Ļāāā,Ļāāā), which I record here:
Lemma 5.19** (Model Operation).**
Let (G,X) be a model for (Ļāā,Ļāā). Assertions: For any model (Gā,Xā) of (Ļāāā,Ļāāā), the underlying graph Gā is obtained from the graph Gāa by introducing one new ā-vertex v and introducing two new edges: edge aāā between vertices āaā and v and edge aāā between āaā and ābā. In particular, if G is connected then: Gā is connected if and only if there is a walk in Gā from āaā to āaā.
Of course, this is just a formalization of Figure 1. Specific information related to embeddings could also be included, but models will only be used in §7 to argue about connectivity via the convenient language of walks, so Lemma 5.19 need not concern itself with embeddings.
Proof.
By the construction of dessins dāfamilles, Proposition 2.6, the ā-vertices (resp. ā-vertices) of the underlying graph correspond to Ļāā-orbits (resp. Ļāā-orbits), edges are elements of E, and an edge eāE is incident to vertices x and y if and only if e is contained in both of the corresponding orbits.
I work directly from Definition 5.1 of ā. Let V be the vertex set of G and Vā that of Gā. Definition 5.1 already defines the edges Eā and its simple relationship to E.
First, define a natural injection Ļ:VāŖVā. Define Ļ on ā-vertices of G as follows: if O is an Ļāā-orbit containing neither a nor b then O is also an Ļāāā-orbit and Ļ(O) is defined to be O again, if O is the Ļāā-orbit containing a then Ļ(O) is defined to be the Ļāāā-orbit containing aāā, and if O is the Ļāā-orbit containing b then Ļ(O) is defined to be the Ļāāā-orbit containing aāā. Note that there is no conflict if a,b represent the same Ļāā-orbit. Similarly, define Ļ on ā-vertices of G as follows: if O is an Ļāā-orbit not containing a then O is also an Ļāāā-orbit and Ļ(O) is defined to be O again, and if O is the Ļāā-orbit containing a then Ļ(O) is defined to be the Ļāāā-orbit containing aāā. Note, by Definition 5.1, that Gā contains only one more vertex: the ā-vertex of aāā, i.e. the vertex corresponding to the Ļāāā-orbit containing aāā, a singleton.
Now, let eāGāa be an edge and let Oāā and Oāā be the Ļāā-orbit and Ļāā-orbit corresponding to its vertices āeā and āeā. By Definition 5.1 and the previous paragraph, e is also contained within the Ļāāā-orbit Ļ(Oāā) and the Ļāāā-orbit Ļ(Oāā).
The previous two paragraphs show that, via Ļ:VāŖVā and Eā{a}āŖEā, Gāa is a subgraph of Gā.
It is immediate from Definition 5.1 that Gā contains only one more vertex than G, the vertex v in the statement of this Lemma: VāāĻ(V) consists of the vertex corresponding to the Ļāāā-orbit containing aāā, a singleton. It is also immediate from Definition 5.1 that Gā contains only two more edges than Gāa: the edges aāā and aāā. It is immediate from the Definition 5.1 and the first paragraph of this proof that aāā and aāā are incident to vertices as described in the statement of this Lemma. This concludes the proof of the first statement.
Now, assume that G is connected. It is trivial that if Gā is connected then there is such a walk. Conversely, suppose that there is such a walk. If Gāa is connected then this is obvious from what was already proved: Gā is constructed from Gāa by attaching edges. So, I can assume that Gāa is disconnected. Necessarily, Gāa=GāāāGāā, where Gāā,Gāā are connected and āaāāGāā, āaāāGāā. To prove that Gā is connected, it is equivalent to prove that if x,yāGā are vertices then there is a walk in Gā from x to y. Since Gā contains only one new vertex, the ā-vertex of edge aāā, I can assume that x,y are vertices of G. Since Gāā,Gāā are connected, I can also assume that xāGāā and yāGāā. But the claim is now obvious: concatenate walks from x to āaā and from āaā to y with the assumed walk from āaā to āaā.
ā
6. Iteration of the Operation
Throughout this section, E is a finite set and (Ļāā,Ļāā) is an arbitrary pair in the symmetric group SEā. Fix distinct a,bāE and let (Ļāāā,Ļāāā) be the reroute of (Ļāā,Ļāā) relative to (a,b).
Definition 6.1**.**
(Ļāāāā,Ļāāāā) is defined to be the reroute of (Ļāāā,Ļāāā) relative to (b,aāā). The natural set Eāā is, By Definition 5.1, Eāā=def{aāā,aāā,bāā,bāā}āEā{a,b}.
Recall that if Ļ,sāSEā then Ļs=defsā Ļā sā1. As promised in §5.1, the following is the relationship of the operation ā to conjugation:
Proposition 6.2**.**
Let tāSEā be the transposition exchanging a and b. Let (Ļāāāā,Ļāāāā) be as in Definition 6.1. Assertions:(Ļātā,Ļāā) is transitive if and only if (Ļāāāā,Ļāāāā) is transitive and the synthetic genus of (Ļātā,Ļāā) is the same as that of (Ļāāāā,Ļāāāā).
Graphically, this is fairly intuitive, but that argument is unrigorous.
Proof.
By Definition 5.1, Ļāāāā is constructed from Ļāā as follows: Insert the symbol aāā immediately before b, insert the symbol bāā immediately before a, replace the symbols a,b by aāā,bāā. Similarly, Ļāāāā is constructed from Ļāā as follows: Replace a by the symbol aāā, replace b by the symbol bāā, introduce the trivial cycles (aāā) and (bāā).
Now, let Ļ:SEāāŖSEāāā be induced by the injection EāŖEāā defined by aā¦aāā, bā¦bāā, xā¦x for all xāEā{a,b}. It is immediate that Ļ(Ļāā)=Ļāāāā. Set Ļāā²ā²ā=defĻ(tā Ļāāā tā1), and note that this is the same as the conjugate of Ļ(Ļāā) by the transposition (aāā,bāā). Set Eā²ā²=defEāāā{aāā,bāā}. Clearly, (Ļāā²ā²ā,Ļāāāā) permutes Eā²ā² and (Ļātā,Ļāā) is transitive if and only if (Ļāā²ā²ā,Ļāāāā) is transitive on Eā²ā². Thus, it is equivalent to show that (Ļāā²ā²ā,Ļāāāā) is transitive if and only if (Ļāāāā,Ļāāāā) is transitive.
It is immediate from the cycle structures of Ļāāāā and Ļāā²ā²ā that if Oā²ā²āEā²ā² is an Ļāā²ā²ā-orbit then there is a unique Ļāāāā-orbit Oāā satisfying Oā²ā²āOāāāOā²ā²āŖ{aāā,bāā}, and all Ļāāāā-orbits are produced in this way. Since aāā,bāā are fixed points of Ļāāāā, it is clear that Oā²ā² is Ļāāāā-stable if and only if Oāā is Ļāāāā-stable.
The first claim now follows from what is possibly the most trivial observation ever to appear in print: A pair (A,B) is transitive if and only if no union of A-orbits is B-stable except ā and E.
The second claim is immediate from the information above: ā£Eāāā£=ā£Eā£+2, O(Ļāāāā)=O(Ļāā)+2, O(Ļāāāā)=O(Ļāā)=O(Ļātā), O(ĻāāāāĻāāāā)=O(ĻātāĻāā) .
ā
Therefore, I need to understand the Type of (Ļāāā,Ļāāā) relative to (b,aāā) in terms of the initial data (Ļāā,Ļāā) and (a,b). I refer to this goal below as ābranchingā. This obviously requires knowledge of Ļāāā(b). The following simple observations, which are immediate from Definition 5.1 of Ļāāā, will be useful:
If Ļāā(b)=b then Ļāāā(b)=aāā.
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If Ļāā(b)=a then Ļāāā(b)=aāā.
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If Ļāā(b)ī =a,b then Ļāāā(b)=Ļāā(b).
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Since Ļāāā(aāā)=b always, Ļāāā(b)ī =b always.
The reader will observe that situations Ļāā(b)=b or Ļāā(b)=a are, after exchanging a and b, precisely those that required special treatment by the proofs in §5. This is expected.
6.1. Type U Branching
By the proof of the Reroute Theorem 5.12, all of aāā,aāā,b represent the same ĻāāāĻāāā-orbit, and the ĻāāāĻāāā-cycle containing them is of the form (ā¦aāāā¦aāāā¦bā¦). As usual, I check the position of Ļāāā(b) relative to b,aāā.
If Ļāā(b)=b then Ļāāā(b)=aāā, so (Ļāāā,Ļāāā) is Type P1. If Ļāā(b)=a then Ļāāā(b)=aāā, so (Ļāāā,Ļāāā) is Type P1. So, I can assume that Ļāā(b)ī =b,a, in which case Ļāāā(b)=Ļāā(b). Note that Ļāā(b)ī =Ļāā(a) also, due to distinctness of a,b.
If Ļāā(b) represents the same ĻāāĻāā-orbit as a then Orbit Transfer Lemma 5.9 (7) says that Ļāāā(b) is contained in the ĻāāāĻāāā-arc from aāā to aāā. Therefore, (Ļāāā,Ļāāā) is Type P1.
If Ļāā(b) represents the same ĻāāĻāā-orbit as Ļāā(a) then Orbit Transfer Lemma 5.9 (8) says that Ļāāā(b) is contained in the ĻāāāĻāāā-arc from aāā to b. Therefore, (Ļāāā,Ļāāā) is Type N.
If Ļāā(b) represents the same ĻāāĻāā-orbit as b then Orbit Transfer Lemma 5.9 (9) says that Ļāāā(b) is contained in the ĻāāāĻāāā-arc from b to aāā. Therefore, (Ļāāā,Ļāāā) is Type P1.
If Ļāā(b) represents a different ĻāāĻāā-orbit than all of a,Ļāā(a),b then it is clear from Lemma 5.5 that Ļāāā(b) represents a different ĻāāāĻāāā-orbit than all of aāā,aāā,b, so (Ļāāā,Ļāāā) is Type P3.
These facts can be summarized:
(1)
If Ļāā(b) represents the same ĻāāĻāā-orbit as a then (Ļāāā,Ļāāā) is Type P1 relative to (b,aāā).
2. (2)
If Ļāā(b) represents the same ĻāāĻāā-orbit as Ļāā(a) then (Ļāāā,Ļāāā) is Type N relative to (b,aāā).
3. (3)
If Ļāā(b) represents the same ĻāāĻāā-orbit as b then (Ļāāā,Ļāāā) is Type P1 relative to (b,aāā).
4. (4)
If Ļāā(b) represents a different ĻāāĻāā-orbit than all of a,Ļāā(a),b then (Ļāāā,Ļāāā) is Type P3 relative to (b,aāā).
6.2. Type N Branching
By the proof of the Reroute Theorem 5.12, no two of aāā,aāā,b represent the same ĻāāāĻāāā-orbit. As usual, I check the position of Ļāāā(b) relative to b,aāā.
If Ļāā(b)=b then Ļāāā(b)=aāā, so (Ļāāā,Ļāāā) is Type U. If Ļāā(b)=a then Ļāāā(b)=aāā, so (Ļāāā,Ļāāā) is Type P4. So, I can assume that Ļāā(b)ī =b,a, in which case Ļāāā(b)=Ļāā(b). Note that Ļāā(b)ī =Ļāā(a) also, due to distinctness of a,b.
If Ļāā(b) is contained in the ĻāāĻāā-arc from a to b then Orbit Transfer Lemma 5.9 (6) says that Ļāāā(b) represents the same ĻāāāĻāāā-orbit as aāā. Therefore, (Ļāāā,Ļāāā) is Type U.
If Ļāā(b) is contained in the ĻāāĻāā-arc from b to Ļāā(a) then Orbit Transfer Lemma 5.9 (4) says that Ļāāā(b) represents the same ĻāāāĻāāā-orbit as b. Therefore, (Ļāāā,Ļāāā) is Type P2.
If Ļāā(b) is contained in the ĻāāĻāā-arc from Ļāā(a) to a then Orbit Transfer Lemma 5.9 (5) says that Ļāāā(b) represents the same ĻāāāĻāāā-orbit as aāā. Therefore, so (Ļāāā,Ļāāā) is Type P4.
If Ļāā(b) represents a different ĻāāĻāā-orbit than all of a,Ļāā(a),b then it is clear from Lemma 5.5 that Ļāāā(b) represents a different ĻāāāĻāāā-orbit than all of aāā,aāā,b, so (Ļāāā,Ļāāā) is Type U.
These facts can be summarized:
(1)
If Ļāā(b) is contained in the ĻāāĻāā-arc from a to b then (Ļāāā,Ļāāā) is Type U relative to (b,aāā).
2. (2)
If Ļāā(b) is contained in the ĻāāĻāā-arc from b to Ļāā(a) then (Ļāāā,Ļāāā) is Type P2 relative to (b,aāā).
3. (3)
If Ļāā(b) is contained in the ĻāāĻāā-arc from Ļāā(a) to a then (Ļāāā,Ļāāā) is Type P4 relative to (b,aāā).
4. (4)
If Ļāā(b) represents a different ĻāāĻāā-orbit than all of a,Ļāā(a),b then (Ļāāā,Ļāāā) is Type U relative to (b,aāā).
For reasons that will be explained in §7, one situation must be separated:
Definition 6.3** (Tame Exceptional #1A).**
(Ļāā,Ļāā) is Tame Exceptional if it is Type N relative to (a,b) and situation (2) occurs.
This is the first of three ātame exceptionalā cases that need to be separated. A more concrete description of this case is: a,Ļāā(a),b,Ļāā(b) represent the same ĻāāĻāā-orbit, their ĻāāĻāā-cycle is of the form (ā¦Ļāā(b)ā¦Ļāā(a)ā¦aā¦bā¦), and Ļāā(b)ī =b (it is allowed that Ļāā(a)=a).
6.3. Type P1 Branching
By the proof of the Reroute Theorem 5.12, all of aāā,aāā,b represent the same ĻāāāĻāāā-orbit, and the ĻāāāĻāāā-cycle containing them is of the form (ā¦aāāā¦aāāā¦bā¦). As usual, I check the position of Ļāāā(b) relative to b,aāā.
If Ļāā(b)=b then Ļāāā(b)=aāā, so (Ļāāā,Ļāāā) is Type N. If Ļāā(b)=a then Ļāāā(b)=aāā, so (Ļāāā,Ļāāā) is Type P1. So, I can assume that Ļāā(b)ī =b,a, in which case Ļāāā(b)=Ļāā(b). Note that Ļāā(b)ī =Ļāā(a) also, due to distinctness of a,b.
If Ļāā(b) is contained in the ĻāāĻāā-arc from a to Ļāā(a) then Orbit Transfer Lemma 5.9 (1) says that Ļāāā(b) is contained in the ĻāāāĻāāā-arc from aāā to b. Therefore, (Ļāāā,Ļāāā) is Type N.
If Ļāā(b) is contained in the ĻāāĻāā-arc from Ļāā(a) to b then Orbit Transfer Lemma 5.9 (2) says that Ļāāā(b) is contained in the ĻāāāĻāāā-arc from aāā to aāā. Therefore, (Ļāāā,Ļāāā) is Type N.
If Ļāā(b) is contained in the ĻāāĻāā-arc from b to a then Orbit Transfer Lemma 5.9 (3) says that Ļāāā(b) is contained in the ĻāāāĻāāā-arc from b to aāā. Therefore, (Ļāāā,Ļāāā) is Type P1.
If Ļāā(b) represents a different ĻāāĻāā-orbit than all of a,Ļāā(a),b then it is clear from Lemma 5.5 that Ļāāā(b) represents a different ĻāāāĻāāā-orbit than all of aāā,aāā,b, so (Ļāāā,Ļāāā) is Type P3.
These facts can be summarized:
(1)
If Ļāā(b) is contained in the ĻāāĻāā-arc from a to Ļāā(a) then (Ļāāā,Ļāāā) is Type N relative to (b,aāā).
2. (2)
If Ļāā(b) is contained in the ĻāāĻāā-arc from Ļāā(a) to b then (Ļāāā,Ļāāā) is Type N relative to (b,aāā).
3. (3)
If Ļāā(b) is contained in the ĻāāĻāā-arc from b to a then (Ļāāā,Ļāāā) is Type P1 relative to (b,aāā).
4. (4)
If Ļāā(b) represents a different ĻāāĻāā-orbit than all of a,Ļāā(a),b then (Ļāāā,Ļāāā) is Type P3 relative to (b,aāā).
It is tempting to merge (1) and (2), but this should not be done since they are fundamentally different. In fact, (2) is the second ātame exceptionalā case that must be separated:
Definition 6.4** (Tame Exceptional #1B).**
(Ļāā,Ļāā) is Tame Exceptional if it is Type P1 relative to (a,b) and situation (2) occurs.
A more concrete description of this case is: a,Ļāā(a),b,Ļāā(b) represent the same ĻāāĻāā-orbit, their ĻāāĻāā-cycle is of the form (ā¦Ļāā(a)ā¦Ļāā(b)ā¦bā¦aā¦), and Ļāā(a)ī =a (it is allowed that Ļāā(b)=b). Note that the two Tame Exceptional cases known so far are exchanged when a and b are exchanged.
6.4. Type P2 Branching
By the proof of the Reroute Theorem 5.12, aāā and b represent the same ĻāāāĻāāā-orbit and aāā represents a different ĻāāāĻāāā-orbit. As usual, I check the position of Ļāāā(b) relative to b,aāā.
If Ļāā(b)=b then Ļāāā(b)=aāā, so (Ļāāā,Ļāāā) is Type P2. If Ļāā(b)=a then Ļāāā(b)=aāā, so (Ļāāā,Ļāāā) is Type P4. So, I can assume that Ļāā(b)ī =b,a, in which case Ļāāā(b)=Ļāā(b). Note that Ļāā(b)ī =Ļāā(a) also, due to distinctness of a,b.
If Ļāā(b) represents the same ĻāāĻāā-orbit as b then Orbit Transfer Lemma 5.9 (9) says that Ļāāā(b) is contained in the ĻāāāĻāāā-arc from b to aāā. Therefore, (Ļāāā,Ļāāā) is Type P2.
If Ļāā(b) represents the same ĻāāĻāā-orbit as a and Ļāā(b) is contained in the ĻāāĻāā-arc from Ļāā(a) to a then Orbit Transfer Lemma 5.9 (5) says that Ļāāā(b) is contained in the ĻāāāĻāāā-orbit of aāā. Therefore, (Ļāāā,Ļāāā) is Type P4.
If Ļāā(b) represents the same ĻāāĻāā-orbit as a but Ļāā(b) is not contained in the ĻāāĻāā-arc from Ļāā(a) to a then Orbit Transfer Lemma 5.9 (1) says that Ļāāā(b) is contained in the ĻāāāĻāāā-arc from aāā to b. Therefore, (Ļāāā,Ļāāā) is Type P2.
If Ļāā(b) represents a different ĻāāĻāā-orbit than all of a,Ļāā(a),b then it is clear from Lemma 5.5 that Ļāāā(b) represents a different ĻāāāĻāāā-orbit than all of aāā,aāā,b. Therefore, (Ļāāā,Ļāāā) is Type U.
These facts can be summarized:
(1)
If Ļāā(b) represents the same ĻāāĻāā-orbit as a and Ļāā(b) is contained in the ĻāāĻāā-arc from Ļāā(a) to a then (Ļāāā,Ļāāā) is Type P4 relative to (b,aāā).
2. (2)
If Ļāā(b) represents the same ĻāāĻāā-orbit as a but Ļāā(b) is not contained in the ĻāāĻāā-arc from Ļāā(a) to a then (Ļāāā,Ļāāā) is Type P2 relative to (b,aāā).
3. (3)
If Ļāā(b) represents the same ĻāāĻāā-orbit as b then (Ļāāā,Ļāāā) is Type P2 relative to (b,aāā).
4. (4)
If Ļāā(b) represents a different ĻāāĻāā-orbit than all of a,Ļāā(a),b then (Ļāāā,Ļāāā) is Type U relative to (b,aāā).
For reasons that will be explained in §7, one situation must be separated:
Definition 6.5** (Wild Exceptional).**
(Ļāā,Ļāā) is Wild Exceptional iff it is Type P2 relative to (a,b) and situation (3) occurs. In other words, iff (Ļāā,Ļāā) is Type P2 relative to both (a,b) and (b,a).
A more concrete description of this case is: a and Ļāā(a) represent the same ĻāāĻāā-orbit, b and Ļāā(b) represent the same ĻāāĻāā-orbit, and the two orbits are different (it is allowed that Ļāā(a)=a and/or Ļāā(b)=b).
6.5. Type P3 Branching
By the proof of the Reroute Theorem 5.12, aāā and b represent the same ĻāāāĻāāā-orbit and aāā represents a different ĻāāāĻāāā-orbit. As usual, I check the position of Ļāāā(b) relative to b,aāā.
If Ļāā(b)=b then Ļāāā(b)=aāā, so (Ļāāā,Ļāāā) is Type P3. If Ļāā(b)=a then Ļāāā(b)=aāā, so (Ļāāā,Ļāāā) is Type P1. So, I can assume that Ļāā(b)ī =b,a, in which case Ļāāā(b)=Ļāā(b). Note that Ļāā(b)ī =Ļāā(a) also, due to distinctness of a,b.
If Ļāā(b) is contained in the ĻāāĻāā-arc from a to b then Orbit Transfer Lemma 5.9 (6) says that Ļāāā(b) represents the same ĻāāāĻāāā-orbit as aāā. Therefore, (Ļāāā,Ļāāā) is Type P3.
If Ļāā(b) is contained in the ĻāāĻāā-arc from b to a then Orbit Transfer Lemma 5.9 (3) says that Ļāāā(b) is contained in the ĻāāāĻāāā-arc from b to aāā. Therefore, (Ļāāā,Ļāāā) is Type P1.
If Ļāā(b) represents the same ĻāāĻāā-orbit as Ļāā(a) then Orbit Transfer Lemma 5.9 (8) says that Ļāāā(b) is contained in the ĻāāāĻāāā-arc from aāā to b. Therefore, (Ļāāā,Ļāāā) is Type N.
If Ļāā(b) represents a different ĻāāĻāā-orbit than all of a,Ļāā(a),b then it is clear from Lemma 5.5 that Ļāāā(b) represents a different ĻāāāĻāāā-orbit than all of aāā,aāā,b. Therefore, (Ļāāā,Ļāāā) is Type P3.
These facts can be summarized:
(1)
If Ļāā(b) is contained in the ĻāāĻāā-arc from a to b then (Ļāāā,Ļāāā) is Type P3 relative to (b,aāā).
2. (2)
If Ļāā(b) is contained in the ĻāāĻāā-arc from b to a then (Ļāāā,Ļāāā) is Type P1 relative to (b,aāā).
3. (3)
If Ļāā(b) represents the same ĻāāĻāā-orbit as Ļāā(a) then (Ļāāā,Ļāāā) is Type N relative to (b,aāā).
4. (4)
If Ļāā(b) represents a different ĻāāĻāā-orbit than all of a,Ļāā(a),b then (Ļāāā,Ļāāā) is Type P3 relative to (b,aāā).
The third and last ātame exceptionalā situation that needs to be separated is:
Definition 6.6** (Tame Exceptional #2).**
(Ļāā,Ļāā) is Tame Exceptional if it is Type P3 relative to (a,b) and situation (3) occurs.
A more concrete description of this case is: a,b represent the same ĻāāĻāā-orbit, Ļāā(a),Ļāā(b) represent the same ĻāāĻāā-orbit, and the two orbits are different.
6.6. Type P4 Branching
By the proof of the Reroute Theorem 5.12, aāā and aāā represent the same ĻāāāĻāāā-orbit and b represents a different ĻāāāĻāāā-orbit. As usual, I check the position of Ļāāā(b) relative to (b,aāā).
If Ļāā(b)=b then Ļāāā(b)=aāā, so (Ļāāā,Ļāāā) is Type P4. If Ļāā(b)=a then Ļāāā(b)=aāā, so (Ļāāā,Ļāāā) is Type P4. So, I can assume that Ļāā(b)ī =b,a, in which case Ļāāā(b)=Ļāā(b). Note that Ļāā(b)ī =Ļāā(a) also, due to distinctness of a,b.
If Ļāā(b) represents the same ĻāāĻāā-orbit as a then Orbit Transfer Lemma 5.9 (7) says that Ļāāā(b) is contained in the ĻāāāĻāāā-arc from aāā to aāā. Therefore, (Ļāāā,Ļāāā) is Type P4.
If Ļāā(b) represents the same ĻāāĻāā-orbit as b and is contained in the ĻāāĻāā-arc from Ļāā(a) to b then Orbit Transfer Lemma 5.9 (2) says that Ļāāā(b) is contained in the ĻāāāĻāāā-arc from aāā to aāā. Therefore, (Ļāāā,Ļāāā) is Type P4.
If Ļāā(b) represents the same ĻāāĻāā-orbit as b and is not contained in the ĻāāĻāā-arc from Ļāā(a) to b then Orbit Transfer Lemma 5.9 (4) says that Ļāāā(b) is contained in the ĻāāāĻāāā-orbit of b. Therefore, (Ļāāā,Ļāāā) is Type P2.
If Ļāā(b) represents a different ĻāāĻāā-orbit than all of a,Ļāā(a),b then it is clear from Lemma 5.5 that Ļāāā(b) represents a different ĻāāāĻāāā-orbit than all of aāā,aāā,b. Therefore, (Ļāāā,Ļāāā) is Type U.
These facts can be summarized:
(1)
If Ļāā(b) represents the same ĻāāĻāā-orbit as a then (Ļāāā,Ļāāā) is Type P4 relative to b,aāā.
2. (2)
If Ļāā(b) represents the same ĻāāĻāā-orbit as b and is contained in the ĻāāĻāā-arc from Ļāā(a) to b then (Ļāāā,Ļāāā) is Type P4 relative to (b,aāā).
3. (3)
If Ļāā(b) represents the same ĻāāĻāā-orbit as b and is not contained in the ĻāāĻāā-arc from Ļāā(a) to b then (Ļāāā,Ļāāā) is Type P2 relative to (b,aāā).
4. (4)
If Ļāā(b) represents a different ĻāāĻāā-orbit than all of a,Ļāā(a),b then (Ļāāā,Ļāāā) is Type U relative to (b,aāā).
7. Classification 1: by Transitivity
Throughout this section, (D,X) is a dessin dāenfant with edges E and monodromy pair (Ļāā,Ļāā). Fix distinct a,bāE. Let (Ļāāā,Ļāāā) be the reroute of (Ļāā,Ļāā) relative to (a,b) and (Ļāāāā,Ļāāāā) as in Definition 6.1. Finally, (Dā,Xā) is a nondegenerate model for (Ļāāā,Ļāāā) and (Dāā,Xāā) is a nondegenerate model for (Ļāāāā,Ļāāāā).
Recall that if Ļ,sāSEā then Ļs=defsā Ļā sā1. In the subsections below, I give an explicit and nearly complete answer to the following question:
For which transpositions tāSEā is (Ļātā,Ļāā) transitive?
I will also show that the exceptional cases in which I do not make any absolute assertion are genuinely ambiguous and equivalent to a more general question that I am currently unable to answer satisfactorily.
7.1. The Non-Exceptional case
For the convenience of the reader, I recall from §6 the concrete descriptions of the exceptional cases:
Definition 7.1**.**
The pair (Ļāā,Ļāā) is
Tame Exceptional #1 relative to (a,b) iff a,Ļāā(a),b,Ļāā(b) are distributed into ĻāāĻāā-cycles as either (ā¦Ļāā(b)ā¦Ļāā(a)ā¦aā¦bā¦) with Ļāā(b)ī =b, or this after exchanging a and b.
-
Tame Exceptional #2 relative to (a,b) iff a,Ļāā(a),b,Ļāā(b) are distributed into ĻāāĻāā-cycles as (ā¦aā¦bā¦),(ā¦Ļāā(a)ā¦Ļāā(b)ā¦).
-
Wild Exceptional relative to (a,b) iff a,Ļāā(a),b,Ļāā(b) are distributed into ĻāāĻāā-cycles as (ā¦aā¦Ļāā(a)ā¦),(ā¦bā¦Ļāā(b)ā¦).
Note that, among Exceptional cases, Wild Exceptional is the only one that preserves synthetic genus ā if (Ļāā,Ļāā) is Tame Exceptional relative to (a,b) then gāā=gā1, for g the synthetic genus of (Ļāā,Ļāā) and gāā that of (Ļāāāā,Ļāāāā).
Transitivity Theorem 7.2**.**
For tāSEā the transposition exchanging a and b, if (Ļāā,Ļāā) is not Exceptional relative to (a,b) then (Ļātā,Ļāā) is transitive.
Analysis of the exceptional cases is given in subsections §7.2, §7.3 below; classification of t according to the genus of (Ļātā,Ļāā) is given in §8.
Proof.
By Proposition 6.2, it suffices to prove that (Ļāāāā,Ļāāāā) is transitive. I heavily use notation of the form āType X(Y)ā, in reference to the analysis of branching given in §6. The proof is by cases.
Many cases can be eliminated by use of the following simple observation: If (Ļāā,Ļāā) is Type U, Type P1, Type P3, or Type P4 relative to (a,b) then (Ļāāā,Ļāāā) is transitive. This is true by Lemma 5.19 and the proof of the Reroute Theorem 5.12: aāā and b represent the same ĻāāāĻāāā-orbit, so there is a walk in Dā from āaā to ābā (cf. §3), and then to āaā along edge aāā. Applying this observation twice, it is seen that (Ļāāāā,Ļāāāā) can only (in principle) be non-transitive if (Ļāā,Ļāā) is Type U(2), Type N(1), Type N(3), Type N(4), Type P1(1), Type P2(1), Type P2(2), Type P2(4), Type P4(3).
More cases can be eliminated by exploiting the symmetry of the claim with respect to a,b: after exchanging a and b, Type N(3) becomes Type P1(1), Type N(4) becomes Type P3(1) (known to be transitive by the previous paragraph), Type P2(1) becomes Type P4(3), Type P2(2) becomes Type P4(2) (also known), Type P2(4) becomes Type U(3) (also known).
Altogether, transitivity must be verified only for Type U(2), Type N(1), Type P1(1), Type P4(3). It is equivalent by Lemma 2.3 to prove that Dāā is connected.
Suppose first that (Ļāā,Ļāā) is Type U(2). In particular, (Ļāāā,Ļāāā) is transitive, so Lemma 5.19 says that it is equivalent to find a walk in Dāā from ābā to ābā. Recall that aāāāDāābāDāāābāā where bāā is the ānewā edge from ābā to āaā. Since aāā connects āaā to ābā, it is therefore sufficient to exhibit a walk in Dāāb from āaā to āaā. By nature of Type U(2), ĻāāāĻāāā contains a cycle of the form (ā¦Ļāāā(b)ā¦bā¦aāāā¦aāāā¦). Let x0ā,x1ā,ā¦,xnāāEā be the minimal ĻāāāĻāāā-sequence from aāā to aāā. As discussed in §3, this produces a walk, via the sequence of edges x0ā,Ļāāā(x0ā),x1ā,ā¦,Ļāāā(xnā1ā),xnā, from either vertex of x0ā to either vertex of xnā. Thus, it remains only to check that this walk exists in Dāāb, i.e. that b is not among the edges x0ā,Ļāāā(x0ā),x1ā,ā¦,Ļāāā(xnā1ā),xnā. Since Ļāāā(Ļāāā(xiā))=xi+1ā for all 0ā¤i<n, it suffices to show that xiāī =b,Ļāāā(b) for all 0ā¤iā¤n. But this is obvious from the structure of the ĻāāāĻāāā-cycle.
Suppose now that (Ļāā,Ļāā) is Type P1(1). This proof is identical to that for Type U(2). Since (Ļāāā,Ļāāā) is transitive, it is equivalent to have a walk in Dāā from ābā to ābā. As before, it is sufficient to exhibit a walk in Dāāb from āaā to āaā. This is done exactly as before: by nature of Type P1(1), ĻāāāĻāāā contains a cycle of the form (ā¦Ļāāā(b)ā¦bā¦aāāā¦aāāā¦), and the walk associated with the minimal ĻāāāĻāāā-sequence from aāā to aāā is contained in Dāāb.
Suppose that (Ļāā,Ļāā) is Type P4(3). The proof is nearly identical to that for Type U(2) and Type P1(1): (Ļāāā,Ļāāā) is transitive and ĻāāāĻāāā contains a cycle of the form (ā¦aāāā¦aāāā¦), with both b and Ļāāā(b) contained in a different cycle, so there is a walk in Dāāb from āaā to āaā.
Suppose finally that (Ļāā,Ļāā) is Type N(1). The proof here is different, because it is possible that (Ļāāā,Ļāāā) is not transitive (so Lemma 5.19 cannot be used). If (Ļāāā,Ļāāā) is transitive then the claim is immediate by the second paragraph of this proof: by nature of Type N(1), (Ļāāā,Ļāāā) is Type U relative to (b,aāā). So, I can assume that Dā is disconnected. Necessarily, Dā=DāāāāDāāā with both Dāāā,Dāāā connected and āaāāDāāā, āaāāDāāā. Since the edge aāāāDā connects āaā to ābā, necessarily ābāāDāāā, which forces bāDāāā and ābāāDāāā. Since the underlying graph of Dāā is formed from Dāāb by attaching a new edge bāā between ābā and āaā, it suffices to show that Dāāāāb is connected. The monodromy pair (Ļāā,Ļāā) of the connected dessin dāfamille (Dāāā,Xā) is simply the restriction of (Ļāāā,Ļāāā) to the edges of Dāāā. In particular, the orbits of ĻāāĻāā are a certain subset of the orbits of ĻāāāĻāāā. By nature of Type N(1), b and Ļāāā(b) represent different ĻāāāĻāāā-orbits. Since, by Corollary 4.2, Dāāāāb disconnected implies that b and Ļāā(b) represent the same ĻāāĻāā-orbit, Dāāāāb must be connected.
ā
Remark*.*
Due to the fact that genus is never negative, the reader may wonder how it is possible that (Ļātā,Ļāā) is always transitive for certain genus-lowering t, specifically Type N(3) and Type P1(1). The answer is that the surface of such a dessin dāenfant is never S. Something of the ātoralā nature of these types can be seen from the ĻāāĻāā-cycle containing a,Ļāā(a),b,Ļāā(b). For example, for Type N(3) there is a ĻāāĻāā-cycle of the form (ā¦aā¦bā¦Ļāā(a)ā¦Ļāā(b)ā¦), which means that the boundary walk of the corresponding face is of the form ā¦ā,a,āā¦ā,b,āā¦ā,a,āā¦ā,b,āā¦, an appearance of the āsquare modelā of T. The same is true for Type P1(1) after exchanging a and b.
Here is a nice corollary in the spherical case:
Corollary 7.3**.**
Assume that X=S and that (Ļāā,Ļāā) is not Wild Exceptional relative to (a,b). Assertion:(Ļātā,Ļāā) is transitive if and only if O(ĻātāĻāā)ā¤O(ĻāāĻāā).
Proof.
If (Ļātā,Ļāā) is transitive and O(ĻātāĻāā)>O(ĻāāĻāā) then the dessin dāenfant corresponding to (Ļātā,Ļāā) has genus strictly lower than that of (Ļāā,Ļāā), which is absurd. Conversely, let gt be the synthetic genus of (Ļātā,Ļāā). By the inner hypothesis, gtā„0. This implies that (Ļāā,Ļāā) is not Tame Exceptional relative to (a,b), since Tame Exceptional would imply gt<0. Combined with the outer hypothesis, (Ļāā,Ļāā) is not Exceptional relative to (a,b). By the Transitivity Theorem 7.2, (Ļātā,Ļāā) is transitive.
ā
If O(ĻātāĻāā)<O(ĻāāĻāā) then (Ļātā,Ļāā) is transitive.
Proof.
By hypothesis, the synthetic genus of (Ļātā,Ļāā) is strictly larger than that of (Ļāā,Ļāā). This implies that (Ļāā,Ļāā) is not Exceptional relative to (a,b) so, by the Transitivity Theorem 7.2, (Ļātā,Ļāā) is transitive.
ā
7.2. The Wild Exceptional case
The following is fairly weak, but still necessary:
Proposition 7.5**.**
Here, (Ļāā,Ļāā) is not necessarily Wild Exceptional relative to (a,b).* Let tāSEā be the transposition exchanging a and b. Let (Dt,Xt) be a model for (Ļātā,Ļāā). Assertion:Dt is connected if and only if at least one of the following exists:*
(1)
a walk in the subgraph Dā(aāŖb) from āaā to āaā
or
2. (2)
a walk in the subgraph Dā(aāŖb) from ābā to ābā
or
3. (3)
a walk in the subgraph Dā(aāŖb) from āaā to ābā
or
4. (4)
a walk in the subgraph Dā(aāŖb) from āaā to ābā
The relation of this to the Wild Exceptional situation is given after the proof.
Proof.
First, note that the number of connected components of Dā(aāŖb) is at most three. Second, each connected component must contain at least one of the (not necessarily distinct) vertices āaā,āaā,ābā,ābā. Third, the underlying graph Dt is constructed from Dā(aāŖb) by attaching two new edges, one between āaā and ābā and another between ābā and āaā (cf. Lemma 5.19 and the proof of Proposition 6.2). Both halves of the equivalence are obvious if Dā(aāŖb) is connected (use the third remark from the beginning of this paragraph). So, I can assume that Dā(aāŖb) is disconnected.
Suppose that Dā(aāŖb) has three connected components. In this case, the second remark can be sharpened: one component C must contain two of āaā,āaā,ābā,ābā and the other two components each contain one. Further, among the six possible pairs, C can only contain one of these four pairs: āaā,ābā or āaā,ābā or āaā,ābā or āaā,ābā. It is clear that the cases in which the attachment of the two new edges unifies the three components into one are these: āaā,ābāāC or āaā,ābāāC. It is also clear that these are precisely the cases among the four in which there is a walk from the list.
Suppose instead that Dā(aāŖb) has two connected components. If one of the two components contains only one of the vertices āaā,āaā,ābā,ābā then both halves of the equivalence are obvious (use the third remarks from the first paragraph). So, I can assume that each connected component contains two of the vertices āaā,āaā,ābā,ābā. By the third remark in the first paragraph, it is clear that the attachment of the two new edges unifies the two components into one precisley when neither component contains both āaā,ābā (equivalently, neither component contains both ābā,āaā). Conversely, this is precisely the situation in which there is a walk from the list.
ā
Remark*.*
If X=S then ā£Ļ0ā(Dā(aāŖb))ā£=3 by Corollary 4.2.
Unfortunately, it seems that no better statement is possible for the Wild Exceptional case. What prohibits the possibility of a better statement, and is the reason for the word āwildā, is the fact that no good relationship need exist between the edges a and b: a and b need not border a common face and may be very far apart within the graph.
In short, understanding of the Wild Exceptional class as a whole requires understanding possibly very long walks from a to b that depend on the global structure of the graph. To the best of my knowledge, not much can be said about this.
By contrast, Tame Exceptional requires that a,b represent the same ĻāāĻāā-orbit; in particular, a and b border a common face and an explicit walk from a to b is easy to construct.
7.3. The Tame Exceptional case
First, I provide simple examples to show the possibilities that can occur.
Example 7.6* (Tame Exceptional #1, spherical / non-transitive).*
Define Ļāā,ĻāāāS3ā by specifying disjoint cycle decompositions: Ļāā=def(1,2)ā (3) and Ļāā=def(1)ā (2,3). This pair (Ļāā,Ļāā) is obviously transitive, ĻāāĻāā=(1,2,3), and the surface of the corresponding dessin dāenfant is S. This pair describes the linear dessin dāenfant with three edges. It is evident that (Ļāā,Ļāā) is Type P1(2) relative to (1,3), and therefore Tame Exceptional #1. For t=(1,3), it is clear that Ļātā=Ļāā and so (Ļātā,Ļāā) is not transitive. Of course, transitivity was expected to fail by considering synthetic genus.
Example 7.7* (Tame Exceptional #1, toral / transitive).*
Define Ļāā,ĻāāāS4ā by specifying disjoint cycle decompositions: Ļāā=def(1,2,3)ā (4) and Ļāā=def(1,2,4,3). This pair (Ļāā,Ļāā) is transitive. It is easy to compute that ĻāāĻāā=(1,3,2,4). In particular, the formula for Euler Characteristic shows that the surface of this dessin dāenfant is T. It is evident that (Ļāā,Ļāā) is Type P1(2) relative to (1,4), and therefore Tame Exceptional #1. The following depicts the dessin dāenfant corresponding to (Ļāā,Ļāā) on the left and, for t=(1,4), a model for (Ļātā,Ļāā) on the right:
Connectedness of the model implies that (Ļātā,Ļāā) is transitive. Note that the ātrueā surface of the dessin dāenfant corresponding to (Ļātā,Ļāā) is S.
Example 7.8* (Tame Exceptional #1, toral / non-transitive).*
This example is created by gluing Example 7.6 onto the trivial dessin dāenfant. Define Ļāā,ĻāāāS6ā by specifying disjoint cycle decompositions: Ļāā=def(1,2)ā (3)ā (4,5,6) and Ļāā=def(1,4,5,6)ā (2,3). The pair (Ļāā,Ļāā) is clearly transitive, ĻāāĻāā=(1,5,4,6,2,3), and the formula for Euler Characteristic shows that the surface of its dessin dāenfant is T. The pair (Ļāā,Ļāā) is Type P1(2) relative to (1,3) and therefore Tame Exceptional #1. For t=(1,3), Ļātā=(3,2)ā (1)ā (4,5,6) and so (Ļātā,Ļāā) is clearly not transitive: both stabilize {2,3}. The following depicts the dessin dāenfant corresponding to (Ļāā,Ļāā) on the left and a (disconnected) model for (Ļātā,Ļāā) on the right:
Example 7.9* (Tame Exceptional #2, spherical / non-transitive).*
Define Ļāā,ĻāāāS4ā by specifying disjoint cycle decompositions: Ļāā=def(1,2)ā (3,4) and Ļāā=def(1,4)ā (2,3). The pair (Ļāā,Ļāā) is clearly transitive. It is easy to compute that ĻāāĻāā=(1,3)ā (2,4). In particular, the formula for Euler Characteristic shows that the surface of its dessin dāenfant is S. This is Example 5.17. It is evident that (Ļāā,Ļāā) is Type P3(3) relative to (1,3), and therefore Tame Exceptional #2. But for t=(1,3), it is clear that Ļātā=Ļāā and Ļāā is not a maximal cycle so (Ļātā,Ļāā) is not transitive. Of course, transitivity was expected to fail by considering synthetic genus.
Example 7.10* (Tame Exceptional #2, toral / transitive).*
Define Ļāā,ĻāāāS8ā by specifying disjoint cycle decompositions: Ļāā=def(1,2,3)ā (4,5,6)ā (7,8) and Ļāā=def(1,7,5)ā (2,6,4)ā (3,8). This pair (Ļāā,Ļāā) is transitive. It is easy to compute that ĻāāĻāā=(1,8)ā (2,4,3,7,6,5). In particular, the formula for Euler Characteristic shows that the surface of this dessin dāenfant is T. It is evident that (Ļāā,Ļāā) is Type P3(3) relative to (1,8), and therefore Tame Exceptional #2. The following depicts the dessin dāenfant corresponding to (Ļāā,Ļāā) on the left and, for t=(1,8), a model for (Ļātā,Ļāā) on the right:
Connectedness of the model implies that (Ļātā,Ļāā) is transitive. Note that the ātrueā surface of the dessin dāenfant corresponding to (Ļātā,Ļāā) is S.
Example 7.11* (Tame Exceptional #2, toral / non-transitive).*
Let (Ļāā,Ļāā) be as in the previous Example 7.10, but choose instead the transposition t=(3,7). It is again true that (Ļāā,Ļāā) is Type P3(3) relative to (3,7), but (Ļātā,Ļāā) is not transitive.
The following shows that to decide transitivity in the Tame Exceptional situation is also equivalent to a seemingly subtle question about deletion vs. connectedness in graphs which, nonetheless, is simpler than that in Proposition 7.5:
Proposition 7.12**.**
Assume that (Ļāā,Ļāā) is Tame Exceptional relative to (a,b). Let tāSEā be the transposition exchanging a and b. Let (Dt,Xt) be a model for (Ļātā,Ļāā). Assertion:Dt is connected if and only if at least one of the following exists:
(1)
a walk in the subgraph Dā(aāŖb) from āaā to āaā
or
2. (2)
a walk in the subgraph Dā(aāŖb) from ābā to ābā
At the end of the subsection, a short discussion of the āwalksā condition is provided.
Proof.
Recall from §7.1 the disjoint cycle decompositions that are possible for Tame Exceptional. Since each half of the equivalence is symmetric in a and b, I can exchange a and b if necessary and assume that the ĻāāĻāā-arc from a to b contains neither Ļāā(a) nor Ļāā(b). If x0ā,x1ā,ā¦,xnāāE is the minimal ĻāāĻāā-sequence from a to b then xiāī =a,b for all 0<i<n and xiāī =Ļāā(a),Ļāā(b) for all 0<iā¤n. This implies that the sequence Ļāā(x0ā),x1ā,ā¦,xnā1ā,Ļāā(xnā1ā) does not contain a,b. As explained in §3, the sequence Ļāā(x0ā),x1ā,ā¦,xnā1ā,Ļāā(xnā1ā) defines a walk W, necessarily in Dā(aāŖb), from āaā to ābā. The equivalence is now immediate from Proposition 7.5: by concatenating with the walk W if necessary, the list of four walks in Proposition 7.5 reduces to the list of two walks here.
ā
Observe that if Dā(aāŖb) is connected then the statement regarding walks in Proposition 7.12 is true, but not conversely. Similarly, if the statement regarding walks in Proposition 7.12 is true then at least one of Dāa or Dāb is connected, but not conversely ā see Example 7.11.
Remark*.*
It is intuitive and tempting to think that the equivalence in Proposition 7.12 might generalize completely, but this is false. On the other hand, it is indeed true generally that if the statement about walks is true then Dt is connected.
The value of Proposition 7.12 would be greatly increased if the following question could be answered:
Deletion Question 2*.*
Is there a āgoodā characterization, in terms of the monodromy pair (Ļāā,Ļāā), of those pairs a,bāE such that there is a walk in Dā(aāŖb) from āaā to āaā or from ābā to ābā?
Since Dāe is connected iff and only if there is a walk in Dāe from āeā to āeā, this is some kind of āsecond orderā analogue of Deletion Question 1 from §4.
8. Classification 2: by Genus
Throughout this section, (G,X) is a dessin dāfamille with edges E and monodromy pair (Ļāā,Ļāā), and g is the synthetic genus of (Ļāā,Ļāā).
Recall that if Ļ,sāSEā then Ļs=defsā Ļā sā1. In this section, I give a complete and explicit answer to the following question:
If tāSEā is a transposition then what is the synthetic genus of (Ļātā,Ļāā)?
This is mostly a matter of collating the facts from §6. Fix a,bāE and let tāSEā be the transposition exchanging a and b.
8.1. Genus-Raising Transpositions
Proposition 8.1**.**
Let gt be the synthetic genus of (Ļātā,Ļāā). Assertions:gt>g if and only if t has the following property:
(1)
At least one of a,b represents a different ĻāāĻāā-orbit than the other three of a,Ļāā(a),b,Ļāā(b)
and
2. (2)
at least one of Ļāā(a),Ļāā(b) represents a different ĻāāĻāā-orbit than the other three of a,Ļāā(a),b,Ļāā(b).
In this case, gt=g+1. If (Ļāā,Ļāā) is transitive then (Ļātā,Ļāā) is also transitive. It is not assumed that a,Ļāā(a),b are distinct.
Proof.
According to §6 and the Reroute Theorem 5.12, the situations in which gt>g are precisely these five: Type U(1), Type U(3), Type U(4), Type P2(4), Type P4(4). The fact that gt=g+1 is then immediate from the Reroute Theorem 5.12. This then implies, in the case that (Ļāā,Ļāā) is transitive, that (Ļātā,Ļāā) is also transitive, by Corollary 7.4. Explicitly, the five situations above are described by saying that a,Ļāā(a),b,Ļāā(b) are distributed into ĻāāĻāā-cycles in one of the following ways:
An easy āplacing balls into boxesā argument shows that the given property extracts precisely these five situations from among all possible.
ā
Proposition 8.1 answers a question asked to me by Brian Hwang, which I record:
Corollary 8.2**.**
Let (D,S) be a spherical dessin dāenfant with edges E and monodromy (Ļāā,Ļāā). Assertion: The transpositions tāSEā for which (Ļātā,Ļāā) defines a dessin dāenfant on T are precisely those satisfying the property in Proposition 8.1.
Remark*.*
There are many dessins dāenfants for which no genus-raising transpositions exist. For example, Proposition 8.1 implies that if a dessin dāenfant has only one face, what Adrianov-Shabat call āunicellularā, then it is impossible for any transposition to be genus-raising.
Remark*.*
Unlike the genus-lowering and genus-preserving transpositions, the genus-raising transpositions can be described using only the concept of āorbitā, rather than the more refined concept of ācycleā.
8.2. Genus-Lowering Transpositions
Proposition 8.3**.**
Let gt be the synthetic genus of (Ļātā,Ļāā). Assertions:gt<g if and only if t has the following property:
(1)
Both a and b represent the same ĻāāĻāā-orbit
and
2. (2)
both Ļāā(a),Ļāā(b) represent the same ĻāāĻāā-orbit
and
3. (3)
either the ĻāāĻāā-arc from a to b contains neither Ļāā(a),Ļāā(b) or the ĻāāĻāā-arc from b to a contains neither.
In this case, gt=gā1. Tame Exceptional situations may occur here, so (Ļātā,Ļāā) may be transitive or not even if (Ļāā,Ļāā) is transitive.
Proof.
According to §6 and the Reroute Theorem 5.12, the situations in which this can occur are precisely Type N(2), Type N(3), Type P1(1), Type P1(2), Type P3(3). The fact that gt=gā1 is then immediate from the Reroute Theorem 5.12. Explicitly, these five cases are described by saying that a,Ļāā(a),b,Ļāā(b) are distributed into ĻāāĻāā-cycles in one of the following ways:
(ā¦Ļāā(b)ā¦Ļāā(a)ā¦aā¦bā¦), with Ļāā(b)ī =b
-
(ā¦Ļāā(a)ā¦Ļāā(b)ā¦aā¦bā¦), with Ļāā(a)ī =b
-
(ā¦Ļāā(b)ā¦Ļāā(a)ā¦bā¦aā¦), with Ļāā(b)ī =a
-
(ā¦Ļāā(a)ā¦Ļāā(b)ā¦bā¦aā¦), with Ļāā(a)ī =a
-
(ā¦aā¦bā¦),(ā¦Ļāā(a)ā¦Ļāā(b)ā¦), with bī =Ļāā(a)
An easy āplacing balls into boxesā argument shows that the given property extracts precisely these five situations from among all possible.
ā
Remark*.*
For some dessins dāenfants, there are no genus-lowering transpositions. This is trivial in S, but examples exist in higher genus. For example, define Ļāā=(1,2,3,4,5) and Ļāā=(1,5,3,2,4). This pair (Ļāā,Ļāā) is transitive, and it is easy to compute that ĻāāĻāā=(1)ā (2,5,4)ā (3), so (Ļāā,Ļāā) defines a dessin dāenfant with synthetic Euler characteristic (1+1)ā5+3=0. Since ĻāāĻāā contains only one non-singleton cycle, the only way that a transposition t=(a,b) can satisfy the first two requirements of Proposition 8.3 is if a,Ļāā(a),b,Ļāā(b) all represent the same ĻāāĻāā-orbit. But it is clear from Ļāā,ĻāāĻāā that there is only one edge e for which e,Ļāā(e) represent the same ĻāāĻāā-orbit: e=4. Therefore, no such a,b can exist.
In particular, some dessins dāenfants in T do not come from S via conjugation by transpositions: If (Ļāā²ā,Ļāā²ā)=(Ļātā,Ļāā²ā) with (Ļāā²ā,Ļāā²ā) toral and (Ļāā,Ļāā²ā) spherical then t must be genus-lowering for (Ļāā²ā,Ļāā²ā).
8.3. Genus-Preserving Transpositions
Although it is possible to describe these directly, it seems best to simply negate those properties from Propositions 8.1 and 8.3.
Appendix: MAGMA
The following MAGMA functions were used to check the assertions in §5, §6, §7. They are quite specific to the goals of this paper, except for one ā the function MakeCycleCoercible should be useful to anyone interested in permutations.
The functions were formatted so they can be processed by MAGMA without editing. In some cases, the definition of the function is preceded by a forward command. This command merely makes explicit that the function depends on some other function defined here.
Format a cycle for coercion
The standard MAGMA function Cycle will, given a permutation and an element, return the part of the permutationās disjoint cycle decomposition containing the element. It is frequently desirable to use this cycle as a permutation. However, the object returned by Cycle is, from MAGMAās perspective, not a permutation at all ā it is merely a sequence of positive integers.
There seems to be no easy or standard way for MAGMA to interpret this sequence as a permutation. For example, an error results if one attempts to coerce (typecast) the sequence into the original symmetric group. It seems that the only way to create a permutation at runtime is to coerce a sequence S, where S[i] indicates the image of i (ātwo row notationā). The following function performs the desired conversion.
The input cycle is an object of type SetIndx (Indexed Set), the same type of object returned by the MAGMA function Cycle. Input n is a positive integer at least as large as the integers in cycle. The returned object is a sequence of n positive integers, can be coerced via Sym(n)! or similar, and the result behaves exactly as cycle should. I deliberately do not coerce the returned object because the user may frequently want to further modify it (I do this myself below, in the body of Reroute).
Compute the synthetic genus of a permutation pair
The following function implements Definition 2.14 and is otherwise self-explanatory.
ComputeGenus := function( white, black )
return ( 1 - ( ( #CycleDecomposition( white ) + #CycleDecomposition( black ) - Degree( Parent( white ) ) + #CycleDecomposition( black*white ) ) / 2 ) );
end function;
There are two other ways to count cycles besides the standard MAGMA function CycleDecomposition: by extracting the quantities of cycles of each length via the standard MAGMA function CycleStructure, or by counting the size of the set returned by the standard MAGMA function Orbits. It is difficult to believe that either of these alternatives is more efficient.
The returned object needs to be explained. Let 1, 2, ā¦, n be the set permuted by the inputs white and black. This function Reroute returns a pair of permutations in the symmetric group on 1, 2, ā¦, n+1. The first one plays the role of Ļāāā and the second one plays the role of Ļāāā. In the new larger symmetric group, the ānewā integer n+1 plays the role of aāā. By abuse of notation, the āoldā integer a plays the role of aāā.
for i := 2 to #aorbit do /a is first element of aorbit/
Append( ~arc, aorbit[i] );
if aorbit[i] eq b then
return arc;
end if;
end for;
end function;
The function ComputeArc does not check whether a and b are in the same g-orbit ā this is the userās responsibility.
Check the Type of a permutation pair
These functions implement Definitions 5.10/5.11 and are all self-explanatory. The implementation of each function reflects as closely as possible the actual mathematical definitions even when this is less economical than a logically equivalent method.
IsTypeU := function( white, black, a, b )
g := black*white; /MAGMA acts on the right!/
aorbit := Cycle( g, a );
wa := a^white;
if ( b in aorbit ) or ( wa in aorbit ) then
return false;
end if;
borbit := Cycle( g, b );
if wa in borbit then
return false;
end if;
return true;
end function;
forward ComputeArc;
IsTypeN := function( white, black, a, b )
aorbit := Cycle( black*white, a ); /MAGMA acts on the right!/
wa := a^white;
if ( b notin aorbit ) or ( wa notin aorbit ) then
return false;
end if;
return wa notin ComputeArc( black*white, a, b );
end function;
forward IsTypeU;
forward IsTypeN;
IsTypeP := function( white, black, a, b )
return not ( IsTypeU(white,black,a,b) or IsTypeN(white,black,a,b) );
end function;
forward ComputeArc;
IsTypeP1 := function( white, black, a, b )
aorbit := Cycle( black*white, a ); /MAGMA acts on the right!/
wa := a^white;
if ( b notin aorbit ) or ( wa notin aorbit ) then
return false;
end if;
return wa in ComputeArc( black*white, a, b );
end function;
IsTypeP2 := function( white, black, a, b )
aorbit := Cycle( black*white, a ); /MAGMA acts on the right!/
return ( ( a^white ) in aorbit ) and ( b notin aorbit );
end function;
IsTypeP3 := function( white, black, a, b )
aorbit := Cycle( black*white, a ); /MAGMA acts on the right!/
return ( b in aorbit ) and ( ( a^white ) notin aorbit );
end function;
IsTypeP4 := function( white, black, a, b )
borbit := Cycle( black*white, b ); /MAGMA acts on the right!/
return ( ( a^white ) in borbit ) and ( a notin borbit );
end function;
Check whether a permutation pair is Exceptional
The following functions test, to varying degrees of specificity, whether a given permutation pair is Exceptional.
IsTameExceptional1B := function( white, black, a, b )
acycle := Cycle( black*white, a ); /MAGMA acts on the right!/
wa := a^white;
wb := b^white;
if (b notin acycle) or (wa notin acycle) or (wb notin acycle) then
return false;
end if;
x := Position( acycle, wa );
y := Position( acycle, wb );
z := Position( acycle, b );
return (1 lt x) and (x lt y) and (y le z); /3rd inequality weak!/
end function;
forward IsTameExceptional1B;
IsTameExceptional1A := function( white, black, a, b )
return IsTameExceptional1B( white, black, b, a );
end function;
IsTameExceptional2 := function( white, black, a, b )
g := black*white; /MAGMA acts on the right!/
wa := a^white;
wb := b^white;
aorbit := Cycle( g, a );
waorbit := Cycle( g, wa );
return (a notin waorbit) and (b in aorbit) and (wb in waorbit);
end function;
forward IsTameExceptional1B;
forward IsTameExceptional1A;
forward IsTameExceptional2;
IsTameExceptional := function( white, black, a, b )
return IsTameExceptional1A( white, black, a, b ) or IsTameExceptional1B( white, black, a, b ) or IsTameExceptional2( white, black, a, b );
end function;
forward IsTypeP2;
IsWildExceptional := function( white, black, a, b )
return IsTypeP2(white,black,a,b) and IsTypeP2(white,black,b,a);
end function;
forward IsTameExceptional;
forward IsWildExceptional;
IsExceptional := function( white, black, a, b )
return IsTameExceptional(white,black,a,b) or IsWildExceptional(white,black,b,a);
end function;
Check how conjugation will change genus
The following functions implement Propositions 8.1/8.3.
IsGenusRaising := function( white, black, a, b )
g := black*white; /MAGMA acts on the right!/
aorbit := Cycle( g, a );
if b in aorbit then
return false;
end if;
wa := a^white;
wb := b^white;
borbit := Cycle( g, b );
if ( ( wa in aorbit ) or ( wb in aorbit ) ) and ( ( wa in borbit ) or ( wb in borbit ) ) then
return false;
end if;
waorbit := Cycle( g, wa );
if wb in waorbit then
return false;
end if;
wborbit := Cycle( g, wb );
if ( ( a in waorbit ) or ( b in waorbit ) ) and ( ( a in wborbit ) or ( b in wborbit ) ) then
return false;
end if;
return true;
end function;
forward ComputeArc;
IsGenusLowering := function( white, black, a, b )
g := black*white; /MAGMA acts on the right!/
aorbit := Cycle( g, a );
if b notin aorbit then
return false;
end if;
wa := a^white;
wb := b^white;
waorbit := Cycle( g, wa );
if wb notin waorbit then
return false;
end if;
arcab := ComputeArc( g, a, b );
arcba := ComputeArc( g, b, a );
return ( ( wa notin arcab ) and ( wb notin arcab ) ) or ( ( wa notin arcba ) and ( wb notin arcba ) );
end function;
forward IsGenusRaising;
forward IsGenusLowering;
IsGenusPreserving := function( white, black, a, b )
return not ( IsGenusRaising( white, black, a, b ) or IsGenusLowering( white, black, a, b ) );