This section lists all functions used for dessins d'enfants
‣ BlackPerm( D ) | ( attribute ) |
Returns: A permutation
This returns the permutation \(\sigma_0\) of the dessin that corresponds to the black vertices.
gap> D := Dessin((1,2,3), (2,3), 3); Dessin((1,2,3), (2,3), 3) gap> BlackPerm(D); (1,2,3)
‣ WhitePerm( D ) | ( attribute ) |
Returns: A permutation
This returns the permutation \(\sigma_1\) of the dessin that corresponds to the white vertices.
gap> D := Dessin((1,2,3), (2,3), 3); Dessin((1,2,3), (2,3) ,3) gap> WhitePerm(D); (2,3)
‣ DegreeDessin( D ) | ( attribute ) |
Returns: A positive integer
This returns the degree of the dessin.
gap> D := Dessin((1,2,3), (2,3), 3); Dessin((1,2,3), (2,3), 3) gap> DegreeDessin(D); 3
‣ Dessin( black, white, d ) | ( operation ) |
Returns: A dessin
This function constructs a dessin object given by monodromy actions. We expect the dessin to be connected, the permutations should act transitively on \(\{1, \dots, d\}\).
gap> D := Dessin((1,2,3), (2,3), 3); Dessin((1,2,3), (2,3), 3)
‣ ValencyList( D ) | ( attribute ) |
Returns: A record
This function returns the valency list of a dessin. It returns a record.
gap> D := Dessin((1,2,3), (2,3), 3); Dessin((1,2,3), (2,3), 3) gap> ValencyList(D); rec( black := [ 3 ], white := [ 1, 2 ] )
‣ Genus( D ) | ( attribute ) |
Returns: A non-negative integer
This function calculates the genus of the dessin (so the genus of the underlying closed Riemann surface).
gap> D := Dessin((1,2,3), (2,3), 3); Dessin((1,2,3), (2,3), 3) gap> Genus(D); 0
‣ IsConnectedDessin( D ) | ( attribute ) |
Returns: A boolean
This checks whether the dessin is connected, so whether the group generated by \(\sigma_0\) and \(\sigma_1\) acts transitively on \(\{1, \dots, \mathrm{deg}(D)\}\).
gap> D := Dessin((1,2,3), (2,3), 3); Dessin((1,2,3), (2,3), 3) gap> IsConnectedDessin(D); true
‣ ConnectedComponentsDessin( D ) | ( function ) |
Returns: A list of lists
This function returns the connected components of the dessin.
gap> D := Dessin((1,2,3), (2,3), 3); Dessin((1,2,3), (2,3), 3) gap> ConnectedComponentsDessin(D); [ Dessin((1,2,3), (2,3), 3) ]
‣ OrigamiGraph( O ) | ( function ) |
Returns: A list
This function returns the stable graph of an origami. It returns a list of lists. The first entry is a list of the genera of the connected components of the dessin of the origami. The second entry is the adjacency matrix of the origami graph.
gap> O := Origami((1,5,6,9,10)(3,4)(7,8)(11,12), (1,3,5,7,9,11)(2,4,10,12,6,8)); Origami((1,5,6,9,10)(3,4)(7,8)(11,12), (1,3,5,7,9,11)(2,4,10,12,6,8), 12) gap> OrigamiGraph(O); [ [ 1, 0 ], [ [ 0, 3 ], [ 2, 0 ] ] ]
‣ DessinOfOrigami( O ) | ( function ) |
Returns: A list of dessins
This function returns a list of the connected components of the dessin associated to the horizontal boundary point of the Teichmüller curve of the origami. Note that the components are "normalized" in the sense that a component of degree \(d\) only uses letters in \(\{1, \dots, d\}\) in the permutations.
gap> O := Origami((1,2,3)(4,5,6), (1,4)(2,5)(3,6)); Origami((1,2,3)(4,5,6), (1,4)(2,5)(3,6), 6) gap> DessinOfOrigami(O); [ Dessin((1,3,2), (1,2,3), 3), Dessin((1,3,2), (1,2,3), 3) ]
‣ AllDessinsOfOrigami( O ) | ( function ) |
Returns: A list of lists of dessins
This function returns a list of the connected components for the dessins associated to the boundary points of the Teichmüller curve of the origami in all possible directions. Note that the components are "normalized" in the sense that a component of degree \(d\) only uses letters in \(\{1, \dots, d\}\) the permutations.
gap> O := Origami((2,3,4), (1,2)); Origami((2,3,4), (1,2), 4) gap> AllDessinsOfOrigami(O); [ [ Dessin((2,4,3), (1,3,4), 4) ], [ Dessin((2,3), (1,3), 3), Dessin((), (), 1) ], [ Dessin((1,4,3,2), (1,4,2,3), 4) ] ]
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