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HyperCells

A GAP package for constructing primitive cells and supercells of hyperbolic lattices

0.9.1-beta

5 March 2024

Patrick M. Lenggenhager
Email: plengg@pks.mpg.de
Homepage: https://patrick-lenggenhager.github.io

Joseph Maciejko
Email: maciejko@ualberta.ca

Tomáš Bzdušek
Email: tomas.bzdusek@uzh.ch

Contents

2 Triangle Groups and Translation Subgroups

2.1 Triangle Groups

  2.1-1 TriangleGroup
  2.1-2 ProperTriangleGroup
  2.1-3 Signature
  2.1-4 FpGroup

2.2 Translation Groups

  2.2-1 TGTranslationGroupFromQuotient
  2.2-2 TGTranslationGroup
  2.2-3 GetProperTriangleGroup
  2.2-4 QuotientHomomorphism
  2.2-5 AsTGSubgroup
  2.2-6 FpGroup
  2.2-7 FpIsomorphism

2.3 Triangle-Group Quotients

  2.3-1 TGQuotient
  2.3-2 TGQuotientName
  2.3-3 TGQuotientGenus
  2.3-4 TriangleGroupSignature
  2.3-5 TGQuotientOrder
  2.3-6 TGQuotientActionType
  2.3-7 TGQuotientRelators
  2.3-8 TGQuotientRelators
  2.3-9 TGQuotientGroup

2.4 Library of Triangle-Group Quotients

  2.4-1 LoadTGQuotients
  2.4-2 ListTriangleGroups
  2.4-3 ListTGQuotients

2.5 Export and Import

2.5-1 ExportString
2.5-2 ImportTGQuotientString

3 Triangle-Group Cells

3.1-1 TGCell
3.1-2 TGCellSymmetric

3.2 Cell Properties

  3.2-1 GetProperTriangleGroup
  3.2-2 CellRelators
  3.2-3 TGCellTranslationGroup
  3.2-4 TGCellPointGroup
  3.2-5 HasTGCellMSWPs
  3.2-6 TGCellMSWPs

3.3 Query Information from Cells

3.3-1 TGCellSchwarzTriangleRep

3.4 Cell Translation Group

  3.4-1 AsTGSubgroup
  3.4-2 Generators
  3.4-3 FpGroup
  3.4-4 FpIsomorphism

3.5 Cell Point Group

  3.5-1 AsQuotientGroup
  3.5-2 AsQuotient
  3.5-3 GetRightTransversal
  3.5-4 QuotientHomomorphism

3.6 Maximally Symmetric Wyckoff Positions

  3.6-1 AsQuotient
  3.6-2 GetRightTransversal
  3.6-3 GetKernel

3.7 Cell Graphs

  3.7-1 TGCellGraph
  3.7-2 GetProperTriangleGroup
  3.7-3 CellRelators
  3.7-4 CellCenter
  3.7-5 CellVertices
  3.7-6 CellVertexPositions
  3.7-7 CellEdges
  3.7-8 CellEdgeTranslations
  3.7-9 CellFaces
  3.7-10 CellFacesWithCenter
  3.7-11 CellBoundary
  3.7-12 GetTGCell
  3.7-13 Exporting TGCellGraph Objects
  3.7-14 Importing TGCellGraph Objects

5.1 Supercell Model Graphs

5.2 Constructing Supercell Models

5.2-1 TGSuperCellModelGraph
5.2-2 RandomTGSuperCellModelGraph

5.3 Supercell Model Properties

  5.3-1 GetProperTriangleGroup
  5.3-2 PrimitiveCellRelators
  5.3-3 SuperCellRelators
  5.3-4 CellCenter
  5.3-5 GetTGPrimitiveCell
  5.3-6 GetTGSuperCell
  5.3-7 GetTGCell
  5.3-8 ModelType
  5.3-9 CellVertices
  5.3-10 CellVertexPositions
  5.3-11 CellEdges
  5.3-12 CellEdgeTranslations
  5.3-13 CellFaces
  5.3-14 CellEmbedding

5.4 Embedding of the Primitive Cell in the Supercell

  5.4-1 PrimitiveCellTranslationGroup
  5.4-2 SuperCellTranslationGroup
  5.4-3 AsQuotient
  5.4-4 GetRightTransversal
  5.4-5 TranslationGroupEmbedding

5.5 Export and Import

5.5-1 Exporting TGSuperCellModelGraph Objects
5.5-2 Importing TGSuperCellModelGraph Objects

6 Sequences of Triangle-Group Quotients

6.1 Check Sequence of Triangle-Group Quotients

6.1-1 IsTGQuotientSequence

6.2 Extending Quotient Sequences

6.2-1 ExtendTGQuotientSequence
6.2-2 NextTGQuotientOptions

6.3 Export and Import

  6.3-1 ExportTGQuotientList
  6.3-2 ExportTGQuotientSequences
  6.3-3 ImportTGQuotientList
  6.3-4 ImportTGQuotientSequences

7.1 HyperCell cell graph (*.hcc)

7.2 HyperCell model graph (*.hcm)

7.3 HyperCell supercell graph (*.hcs)

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