This chapter introduces methods to find all simplicial surfaces with certain restrictions.
This section contains the methods to compute all essential simplicial discs.
‣ IsEssentialDisc( disc ) | ( operation ) |
Returns: true or false
This function returns true if disc is an essential disc and false else. A simplicial surface disc is called an essential disc if disc is a connected simplicial surfaces of Euler characteristic 1 and every circular vertex-edge path of length 3 bounds a face of disc, and every boundary vertex has degree at least 2 and and no inner edge joins two boundary vertices.
gap> disc := SimplicialSurfaceByDownwardIncidence( > [ [ 1, 3 ], [ 1, 2 ], [ 2, 3 ], [ 3, 4 ], [ 1, 4 ], [ 1, 5 ], [ 1, 6 ], > [ 2, 6 ], [ 2, 7 ], [ 3, 7 ], [ 4, 7 ], [ 4, 8 ], [ 4, 5 ], [ 5, 6 ], > [ 6, 7 ], [ 7, 8 ], [ 5, 8 ] ], > [ [ 1, 2, 3 ], [ 1, 4, 5 ], [ 6, 7, 14 ], [ 2, 7, 8 ], [ 8, 9, 15 ], > [ 3, 9, 10 ], [ 4, 10, 11 ], [ 11, 12, 16 ], [ 12, 13, 17 ], > [ 5, 6, 13 ] ] ); simplicial surface (8 vertices, 17 edges, and 10 faces) gap> IsConnectedSurface(disc); true gap> EulerCharacteristic(disc); 1 gap> CounterOfVertices(disc); counter of vertices ([ 2, 3, 4, 5 ] degrees, and [ 1, 2, 3, 2 ] multiplicities) gap> IsEssentialDisc(disc); true
‣ AllSimplicialSurfacesByEssentialButterflyInsertion( surface ) | ( operation ) |
Returns: a list of simplicial surfaces
This function computes representatives of the isomorphism classes of all simplicial surfaces that can be obtained from the input surface surf by an essential butterfly insertion. An essential butterfly insertion is a butterfly insertion that does not create a 3-waist.
gap> surface := SimplicialSurfaceByDownwardIncidence( > [ [ 1, 3 ], [ 1, 2 ], [ 2, 3 ], [ 3, 4 ], [ 1, 4 ], [ 1, 5 ], [ 1, 6 ], > [ 2, 6 ], [ 2, 7 ], [ 3, 7 ], [ 3, 8 ], [ 4, 8 ], [ 4, 5 ], [ 5, 6 ], > [ 6, 7 ], [ 7, 8 ], [ 5, 8 ] ], > [ [1, 2, 3], [1, 4, 5], [6, 7, 14], [2, 7, 8], [8, 9, 15], > [3, 9, 10], [10, 11, 16], [4, 11, 12], [12, 13, 17], [5, 6, 13] ]); simplicial surface (8 vertices, 17 edges, and 10 faces) gap> IsConnectedSurface(surface); true gap> EulerCharacteristic(surface); 1 gap> AllSimplicialSurfacesByEssentialButterflyInsertion(surface); [ simplicial surface (9 vertices, 20 edges, and 12 faces), simplicial surface (9 vertices, 20 edges, and 12 faces) , simplicial surface (9 vertices, 20 edges, and 12 faces), simplicial surface (9 vertices, 20 edges, and 12 faces) ]
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