GAP (Origami) - Chapter 4: The Origami Database

To use the origami database, you need an instance of ArangoDB running on port 8529. After installing ArangoDB, you need to set up a database as follows:

In this database, create three collections called origami_representatives, origamis and veechgroups. Please download the data from https://www.math.uni-sb.de/ag/weitze/CMS/index.php/de/forschung and import the json-files into the corresponding collections. All of this can be done in the web interface of ArangoDB, which you can reach under http://127.0.0.1:8529.

4.2 Using the origami database

4.2-1 ConnectToOrigamiDB

‣ ConnectToOrigamiDB( ) ( function )

Connects to the arango database 'origami' at the endpoint 'http+tcp://127.0.0.1:8529' with the user 'origami' and password 'secret'.

‣ InsertVeechGroupIntoDB( G ) ( operation )

Takes a ModularSubgroup G and inserts it into the Veech group table. Only precomputed attributes are inserted. This function does not compute anything new from the group. No test is performed to check whether G already exists in the database. The resulting ArangoDocument corresponding to the new database entry is returned.

‣ GetVeechGroupDBEntry( G ) ( operation )

Takes a ModularSubgroup G and returns the corresponding database entry or fail if G is not in the database.

‣ GetVeechGroupsFromDB( constraints ) ( operation )

Returns all Veech groups in the database subject to the argument constraints which is given in the form of a record.

      
      gap> GetVeechGroupsFromDB(rec(index := 30, genus := 0));
      [list of modular subgroups]
      gap> GetVeechGroupsFromDB(rec(congruence := true, level := 12));
      [list of modular subgroups]
      gap> GetVeechGroupsFromDB(rec(deficiency := 5));
      [list of modular subgroups]

‣ UpdateVeechGroupDBEntry( G ) ( operation )

Updates the Veech group entry in the database with newly computed data and returns the corresponding ArangoDocument.

‣ RemoveVeechGroupFromDB( G ) ( operation )

‣ InsertOrigamiRepresentativeIntoDB( O ) ( operation )

Inserts the normal form of the origami O into the origami representative list. Inserts only precomputed data. If the Veech group of O does not already exist in the database, it is also inserted. This function assumes that no other element of the SL_2(Z) orbit of O exists in the representative list, no test is performed to check this.

‣ GetOrigamiOrbitRepresentativeDBEntry( O ) ( operation )

Takes an Origami O and returns the corresponding database entry or fail if O is not in the database.

‣ GetOrigamiOrbitRepresentativesFromDB( constraints ) ( operation )

Returns all origami orbit representatives in the database subject to the constraints given in the form of the record constraints.

      
      gap> GetOrigamiOrbitRepresentativesFromDB(rec(stratum := [4]));
      [ Origami((1,4,3)(2,5), (1,5,3,4)(2,6), 6) ]
      gap> GetOrigamiOrbitRepresentativesFromDB(rec(degree := 5));
      [ Origami((1,3,5), (1,3)(2,4,5), 5) ]

‣ GetAllOrigamiOrbitRepresentativesFromDB( ) ( operation )

Returns all origami orbit representatives in the database. Shorthand for GetOrigamiOrbitRepresentativesFromDB(rec());

‣ UpdateOrigamiOrbitRepresentativeDBEntry( O ) ( operation )

Updates the origami orbit representative entry in the database with newly computed data and returns the corresponding ArangoDocument. If the origami orbit representative O does not exist in the database, it is inserted.

‣ RemoveOrigamiOrbitRepresentativeFromDB( O ) ( operation )

Removes the origami orbit representative O from the database if present. NOT YET IMPLEMENTED!

‣ InsertOrigamiWithOrbitRepresentativeIntoDB( O, R ) ( operation )

Inserts the origami O into the origami list with orbit representative R which is inserted into the origami orbit representative list if it is not already in there. Returns the ArangoDocument corresponding to O.

‣ InsertOrigamiIntoDB( O ) ( operation )

Inserts an origami O into the database. If the Veech group and the orbit of O is known, this function checks if there is already a representative of the orbit of O in the database. If not, O is inserted as the representative of its orbit, if yes, O is only inserted into the origami list with a reference to the representative. If the Veech group of O is not known, it is inserted as its own representative. This might result in entries in the origami orbit representative list which are in the same orbit.

‣ GetOrigamiDBEntry( O ) ( operation )

Returns the ArangoDocument in the origami list corresponding to the origami O or fail if does not exist in the database.

‣ GetOrigamiOrbit( O ) ( operation )

‣ UpdateRepresentativeOfOrigami( O, R ) ( operation )

‣ RemoveOrigamiFromDB( O ) ( operation )

4 The Origami Database

4.1 Setting up the database

4.2 Using the origami database

4.2-1 ConnectToOrigamiDB