  
  
                        [1XHomological Algebra Programming[101X
  
  
                                      [1XHAP[101X
  
  
                                  Version 1.74
  
  
                                10 January 2026
  
  
                                  Graham Ellis
  
  
  
  Graham Ellis
      Email:    [7Xmailto:Graham.Ellis@nuigalway.ie[107X
  
  
  Address: [33X[0;9YSchool of Mathematics[133X
           [33X[0;9YNational University of Irelnd, Galway[133X
           [33X[0;9YGalway[133X
           [33X[0;9Y(Ireland)[133X
  
  
  -------------------------------------------------------
  [1XAbstract[101X
  [33X[0;0Y[12XHAP[112X  is  a homological algebra library for use with the GAP computer algebra
  system,  and  is  still  under  development.  The  current  version 1.74 was
  released on 10 January 2026 .[133X
  [33X[0;0YThe  initial  focus  of  the  library  was  on  computations  related to the
  cohomology  of  finite  and  infinite  groups,  with  particular emphasis on
  integral  coefficients.  The  focus  has since broadened to include Steenrod
  algebras of finite groups, Bredon homology, cohomology of simplicial groups,
  and   general   computations   in  algebraic  topology  relating  to  finite
  CW-complexes,  covering  spaces, knots, knotted surfaces, and topics such as
  persitent homology arising in topological data analysis.[133X
  [33X[0;0YThis   document   describes   the   functions  available  in  [12XHAP[112X.  Examples
  illustrating   these   functions   are   available   in   the  [12XHAP[112X  tutorial
  ([7X../tutorial/chap0.html[107X).[133X
  
  
  -------------------------------------------------------
  [1XCopyright[101X
  [33X[0;0Y© 2005-2019 by Graham Ellis[133X
  
  
  -------------------------------------------------------
  [1XAcknowledgements[101X
  [33X[0;0YThe  HAP  project  has  been supported by the School of Maths at NUI Galway,
  various  PhD  students and postdoctoral researchers at NUI Galway, the Irish
  Research  Council,  Science  Foundation  Ireland,  and  the  EU  Marie Curie
  programme.[133X
  
  
  -------------------------------------------------------
  
  
  [1XContents (HAP commands)[101X
  
  1 [33X[0;0YBasic functionality for cellular complexes, fundamental groups and
  homology[133X
    1.1 [33X[0;0YData [22X⟶[122X Cellular Complexes[133X
      1.1-1 RegularCWPolytope
      1.1-2 CubicalComplex
      1.1-3 PureCubicalComplex
      1.1-4 PureCubicalKnot
      1.1-5 PurePermutahedralKnot
      1.1-6 PurePermutahedralComplex
      1.1-7 CayleyGraphOfGroup
      1.1-8 EquivariantEuclideanSpace
      1.1-9 EquivariantOrbitPolytope
      1.1-10 EquivariantTwoComplex
      1.1-11 QuillenComplex
      1.1-12 RestrictedEquivariantCWComplex
      1.1-13 RandomSimplicialGraph
      1.1-14 RandomSimplicialTwoComplex
      1.1-15 ReadCSVfileAsPureCubicalKnot
      1.1-16 ReadImageAsPureCubicalComplex
      1.1-17 ReadImageAsFilteredPureCubicalComplex
      1.1-18 ReadImageAsWeightFunction
      1.1-19 ReadPDBfileAsPureCubicalComplex
      1.1-20 ReadPDBfileAsPurepermutahedralComplex
      1.1-21 RegularCWPolytope
      1.1-22 SimplicialComplex
      1.1-23 SymmetricMatrixToFilteredGraph
      1.1-24 SymmetricMatrixToGraph
    1.2 [33X[0;0YMetric Spaces[133X
      1.2-1 CayleyMetric
      1.2-2 EuclideanMetric
      1.2-3 EuclideanSquaredMetric
      1.2-4 HammingMetric
      1.2-5 KendallMetric
      1.2-6 ManhattanMetric
      1.2-7 VectorsToSymmetricMatrix
    1.3 [33X[0;0YCellular Complexes [22X⟶[122X Cellular Complexes[133X
      1.3-1 BoundaryMap
      1.3-2 CliqueComplex
      1.3-3 ConcentricFiltration
      1.3-4 DirectProduct
      1.3-5 FiltrationTerm
      1.3-6 Graph
      1.3-7 HomotopyGraph
      1.3-8 Nerve
      1.3-9 RegularCWComplex
      1.3-10 RegularCWMap
      1.3-11 ThickeningFiltration
    1.4 [33X[0;0YCellular Complexes [22X⟶[122X Cellular Complexes (Preserving Data Types)[133X
      1.4-1 ContractedComplex
      1.4-2 ContractibleSubcomplex
      1.4-3 KnotReflection
      1.4-4 KnotSum
      1.4-5 OrientRegularCWComplex
      1.4-6 PathComponent
      1.4-7 PureComplexBoundary
      1.4-8 PureComplexComplement
      1.4-9 PureComplexDifference
      1.4-10 PureComplexInterstection
      1.4-11 PureComplexThickened
      1.4-12 PureComplexUnion
      1.4-13 SimplifiedComplex
      1.4-14 ZigZagContractedComplex
    1.5 [33X[0;0YCellular Complexes [22X⟶[122X Homotopy Invariants[133X
      1.5-1 AlexanderPolynomial
      1.5-2 BettiNumber
      1.5-3 EulerCharacteristic
      1.5-4 EulerIntegral
      1.5-5 FundamentalGroup
      1.5-6 FundamentalGroupOfQuotient
      1.5-7 IsAspherical
      1.5-8 KnotGroup
      1.5-9 PiZero
      1.5-10 PersistentBettiNumbers
    1.6 [33X[0;0YData [22X⟶[122X Homotopy Invariants[133X
      1.6-1 DendrogramMat
    1.7 [33X[0;0YCellular Complexes [22X⟶[122X Non Homotopy Invariants[133X
      1.7-1 ChainComplex
      1.7-2 ChainComplexEquivalence
      1.7-3 ChainComplexOfQuotient
      1.7-4 ChainMap
      1.7-5 CochainComplex
      1.7-6 CriticalCells
      1.7-7 DiagonalApproximation
      1.7-8 Size
    1.8 [33X[0;0Y(Co)chain Complexes [22X⟶[122X (Co)chain Complexes[133X
      1.8-1 FilteredTensorWithIntegers
      1.8-2 FilteredTensorWithIntegersModP
      1.8-3 HomToIntegers
      1.8-4 TensorWithIntegersModP
    1.9 [33X[0;0Y(Co)chain Complexes [22X⟶[122X Homotopy Invariants[133X
      1.9-1 Cohomology
      1.9-2 CupProduct
      1.9-3 Homology
    1.10 [33X[0;0YVisualization[133X
      1.10-1 BarCodeDisplay
      1.10-2 BarCodeCompactDisplay
      1.10-3 CayleyGraphOfGroup
      1.10-4 Display
      1.10-5 DisplayArcPresentation
      1.10-6 DisplayCSVKnotFile
      1.10-7 DisplayDendrogram
      1.10-8 DisplayDendrogramMat
      1.10-9 DisplayPDBfile
      1.10-10 OrbitPolytope
      1.10-11 ScatterPlot
  2 [33X[0;0YBasic functionality for [22XZG[122X-resolutions and group cohomology[133X
    2.1 [33X[0;0YResolutions[133X
      2.1-1 EquivariantChainMap
      2.1-2 FreeGResolution
      2.1-3 ResolutionBieberbachGroup
      2.1-4 ResolutionCubicalCrystGroup
      2.1-5 ResolutionFiniteGroup
      2.1-6 ResolutionNilpotentGroup
      2.1-7 ResolutionNormalSeries
      2.1-8 ResolutionPrimePowerGroup
      2.1-9 ResolutionSL2Z
      2.1-10 ResolutionSmallGroup
      2.1-11 ResolutionSubgroup
    2.2 [33X[0;0YAlgebras [22X⟶[122X (Co)chain Complexes[133X
      2.2-1 LeibnizComplex
    2.3 [33X[0;0YResolutions [22X⟶[122X (Co)chain Complexes[133X
      2.3-1 HomToIntegers
      2.3-2 HomToIntegralModule
      2.3-3 TensorWithIntegers
      2.3-4 TensorWithIntegersModP
    2.4 [33X[0;0YCohomology rings[133X
      2.4-1 AreIsomorphicGradedAlgebras
      2.4-2 HAPDerivation
      2.4-3 HilbertPoincareSeries
      2.4-4 HomologyOfDerivation
      2.4-5 IntegralCohomologyGenerators
      2.4-6 LHSSpectralSequence
      2.4-7 LHSSpectralSequenceLastSheet
      2.4-8 ModPCohomologyGenerators
      2.4-9 ModPCohomologyRing
      2.4-10 Mod2CohomologyRingPresentation
    2.5 [33X[0;0YGroup Invariants[133X
      2.5-1 GroupCohomology
      2.5-2 GroupHomology
      2.5-3 PrimePartDerivedFunctor
      2.5-4 PoincareSeries
      2.5-5 PoincareSeries
      2.5-6 RankHomologyPGroup
    2.6 [33X[0;0Y[22XF_p[122X-modules[133X
      2.6-1 GroupAlgebraAsFpGModule
      2.6-2 Radical
      2.6-3 RadicalSeries
  3 [33X[0;0YBasic functionality for homological group theory[133X
    3.1 [33X[0;0YCocycles[133X
      3.1-1 CcGroup
      3.1-2 CocycleCondition
      3.1-3 StandardCocycle
    3.2 [33X[0;0YG-Outer Groups[133X
      3.2-1 ActedGroup
      3.2-2 ActingGroup
      3.2-3 Centre
      3.2-4 GOuterGroup
    3.3 [33X[0;0Y[22XG[122X-cocomplexes[133X
      3.3-1 CohomologyModule
      3.3-2 HomToGModule
  4 [33X[0;0YBasic functionality for parallel computation[133X
    4.1 [33X[0;0YSix Core Functions[133X
      4.1-1 ChildCreate
      4.1-2 ChildCreate
  5 [33X[0;0YResolutions of the ground ring[133X
    5.1 [33X[0;0Y [133X
      5.1-1 TietzeReducedResolution
      5.1-2 ResolutionArithmeticGroup
      5.1-3 FreeGResolution
      5.1-4 ResolutionGTree
      5.1-5 ResolutionSL2Z
      5.1-6 ResolutionAbelianGroup
      5.1-7 ResolutionAlmostCrystalGroup
      5.1-8 ResolutionAlmostCrystalQuotient
      5.1-9 ResolutionArtinGroup
      5.1-10 ResolutionAsphericalPresentation
      5.1-11 ResolutionBieberbachGroup
      5.1-12 ResolutionCoxeterGroup
      5.1-13 ResolutionDirectProduct
      5.1-14 ResolutionExtension
      5.1-15 ResolutionFiniteDirectProduct
      5.1-16 ResolutionFiniteExtension
      5.1-17 ResolutionFiniteGroup
      5.1-18 ResolutionFiniteSubgroup
      5.1-19 ResolutionGraphOfGroups
      5.1-20 ResolutionNilpotentGroup
      5.1-21 ResolutionNormalSeries
      5.1-22 ResolutionPrimePowerGroup
      5.1-23 ResolutionSmallFpGroup
      5.1-24 ResolutionSubgroup
      5.1-25 ResolutionSubnormalSeries
      5.1-26 TwistedTensorProduct
      5.1-27 ConjugatedResolution
      5.1-28 RecalculateIncidenceNumbers
  6 [33X[0;0YResolutions of modules[133X
    6.1 [33X[0;0Y [133X
      6.1-1 ResolutionFpGModule
  7 [33X[0;0YInduced equivariant chain maps[133X
    7.1 [33X[0;0Y [133X
      7.1-1 EquivariantChainMap
  8 [33X[0;0YFunctors[133X
    8.1 [33X[0;0Y [133X
      8.1-1 ExtendScalars
      8.1-2 HomToIntegers
      8.1-3 HomToIntegersModP
      8.1-4 HomToIntegralModule
      8.1-5 TensorWithIntegralModule
      8.1-6 HomToGModule
      8.1-7 InduceScalars
      8.1-8 LowerCentralSeriesLieAlgebra
      8.1-9 TensorWithIntegers
      8.1-10 FilteredTensorWithIntegers
      8.1-11 TensorWithTwistedIntegers
      8.1-12 TensorWithIntegersModP
      8.1-13 TensorWithTwistedIntegersModP
      8.1-14 TensorWithRationals
  9 [33X[0;0YChain complexes[133X
    9.1 [33X[0;0Y [133X
      9.1-1 ChainComplex
      9.1-2 ChainComplexOfPair
      9.1-3 ChevalleyEilenbergComplex
      9.1-4 LeibnizComplex
      9.1-5 SuspendedChainComplex
      9.1-6 ReducedSuspendedChainComplex
      9.1-7 CoreducedChainComplex
      9.1-8 TensorProductOfChainComplexes
      9.1-9 LefschetzNumber
  10 [33X[0;0YSparse Chain complexes[133X
    10.1 [33X[0;0Y [133X
      10.1-1 SparseMat
      10.1-2 TransposeOfSparseMat
      10.1-3 ReverseSparseMat
      10.1-4 SparseRowMult
      10.1-5 SparseRowInterchange
      10.1-6 SparseRowAdd
      10.1-7 SparseSemiEchelon
      10.1-8 RankMatDestructive
      10.1-9 RankMat
      10.1-10 SparseChainComplex
      10.1-11 SparseChainComplexOfRegularCWComplex
      10.1-12 SparseBoundaryMatrix
      10.1-13 Bettinumbers
  11 [33X[0;0YHomology and cohomology groups[133X
    11.1 [33X[0;0Y [133X
      11.1-1 Cohomology
      11.1-2 CohomologyModule
      11.1-3 CohomologyPrimePart
      11.1-4 GroupCohomology
      11.1-5 GroupHomology
      11.1-6 PersistentHomologyOfQuotientGroupSeries
      11.1-7 PersistentCohomologyOfQuotientGroupSeries
      11.1-8 UniversalBarCode
      11.1-9 PersistentHomologyOfSubGroupSeries
      11.1-10 PersistentHomologyOfFilteredChainComplex
      11.1-11 PersistentHomologyOfCommutativeDiagramOfPGroups
      11.1-12 PersistentHomologyOfFilteredPureCubicalComplex
      11.1-13 PersistentHomologyOfPureCubicalComplex
      11.1-14 ZZPersistentHomologyOfPureCubicalComplex
      11.1-15 RipsHomology
      11.1-16 BarCode
      11.1-17 BarCodeDisplay
      11.1-18 Homology
      11.1-19 HomologyPb
      11.1-20 HomologyVectorSpace
      11.1-21 HomologyPrimePart
      11.1-22 LeibnizAlgebraHomology
      11.1-23 LieAlgebraHomology
      11.1-24 PrimePartDerivedFunctor
      11.1-25 RankHomologyPGroup
      11.1-26 RankPrimeHomology
  12 [33X[0;0YPoincare series[133X
    12.1 [33X[0;0Y [133X
      12.1-1 EfficientNormalSubgroups
      12.1-2 ExpansionOfRationalFunction
      12.1-3 PoincareSeries
      12.1-4 PoincareSeriesPrimePart
      12.1-5 PoincareSeriesLHS
      12.1-6 Prank
  13 [33X[0;0YCohomology ring structure[133X
    13.1 [33X[0;0Y [133X
      13.1-1 IntegralCupProduct
      13.1-2 IntegralRingGenerators
      13.1-3 ModPCohomologyGenerators
      13.1-4 ModPCohomologyRing
      13.1-5 ModPRingGenerators
      13.1-6 Mod2CohomologyRingPresentation
  14 [33X[0;0YCohomology rings of [22Xp[122X-groups (mainly [22Xp=2)[122X[133X
    14.1 [33X[0;0Y [133X
      14.1-1 Mod2CohomologyRingPresentation
      14.1-2 PoincareSeriesLHS
  15 [33X[0;0YCommutator and nonabelian tensor computations[133X
    15.1 [33X[0;0Y [133X
      15.1-1 BaerInvariant
      15.1-2 BogomolovMultiplier
      15.1-3 Bogomology
      15.1-4 Coclass
      15.1-5 EpiCentre
      15.1-6 NonabelianExteriorProduct
      15.1-7 NonabelianSymmetricKernel
      15.1-8 NonabelianSymmetricSquare
      15.1-9 NonabelianTensorProduct
      15.1-10 NonabelianTensorSquare
      15.1-11 RelativeSchurMultiplier
      15.1-12 TensorCentre
      15.1-13 ThirdHomotopyGroupOfSuspensionB
      15.1-14 UpperEpicentralSeries
  16 [33X[0;0YLie commutators and nonabelian Lie tensors[133X
    16.1 [33X[0;0Y [133X
      16.1-1 LieCoveringHomomorphism
      16.1-2 LeibnizQuasiCoveringHomomorphism
      16.1-3 LieEpiCentre
      16.1-4 LieExteriorSquare
      16.1-5 LieTensorSquare
      16.1-6 LieTensorCentre
  17 [33X[0;0YGenerators and relators of groups[133X
    17.1 [33X[0;0Y [133X
      17.1-1 CayleyGraphOfGroupDisplay
      17.1-2 IdentityAmongRelatorsDisplay
      17.1-3 IsAspherical
      17.1-4 PresentationOfResolution
      17.1-5 TorsionGeneratorsAbelianGroup
  18 [33X[0;0YOrbit polytopes and fundamental domains[133X
    18.1 [33X[0;0Y [133X
      18.1-1 CoxeterComplex
      18.1-2 ContractibleGcomplex
      18.1-3 QuotientOfContractibleGcomplex
      18.1-4 TruncatedGComplex
      18.1-5 FundamentalDomainStandardSpaceGroup
      18.1-6 OrbitPolytope
      18.1-7 PolytopalComplex
      18.1-8 PolytopalGenerators
      18.1-9 VectorStabilizer
  19 [33X[0;0YCocycles[133X
    19.1 [33X[0;0Y [133X
      19.1-1 CcGroup
      19.1-2 CocycleCondition
      19.1-3 StandardCocycle
      19.1-4 Syzygy
  20 [33X[0;0YWords in free [22XZG[122X-modules[133X
    20.1 [33X[0;0Y [133X
      20.1-1 AddFreeWords
      20.1-2 AddFreeWordsModP
      20.1-3 AlgebraicReduction
      20.1-4 MultiplyWord
      20.1-5 Negate
      20.1-6 NegateWord
      20.1-7 PrintZGword
      20.1-8 TietzeReduction
      20.1-9 ResolutionBoundaryOfWord
  21 [33X[0;0Y[22XFpG[122X-modules[133X
    21.1 [33X[0;0Y [133X
      21.1-1 CompositionSeriesOfFpGModules
      21.1-2 DirectSumOfFpGModules
      21.1-3 FpGModule
      21.1-4 FpGModuleDualBasis
      21.1-5 FpGModuleHomomorphism
      21.1-6 DesuspensionFpGModule
      21.1-7 RadicalOfFpGModule
      21.1-8 RadicalSeriesOfFpGModule
      21.1-9 GeneratorsOfFpGModule
      21.1-10 ImageOfFpGModuleHomomorphism
      21.1-11 GroupAlgebraAsFpGModule
      21.1-12 IntersectionOfFpGModules
      21.1-13 IsFpGModuleHomomorphismData
      21.1-14 MaximalSubmoduleOfFpGModule
      21.1-15 MaximalSubmodulesOfFpGModule
      21.1-16 MultipleOfFpGModule
      21.1-17 ProjectedFpGModule
      21.1-18 RandomHomomorphismOfFpGModules
      21.1-19 Rank
      21.1-20 SumOfFpGModules
      21.1-21 SumOp
      21.1-22 VectorsToFpGModuleWords
  22 [33X[0;0YMeataxe modules[133X
    22.1 [33X[0;0Y [133X
      22.1-1 DesuspensionMtxModule
      22.1-2 FpG_to_MtxModule
      22.1-3 GeneratorsOfMtxModule
  23 [33X[0;0YG-Outer Groups[133X
    23.1 [33X[0;0Y [133X
      23.1-1 GOuterGroup
      23.1-2 GOuterGroupHomomorphismNC
      23.1-3 GOuterHomomorphismTester
      23.1-4 Centre
      23.1-5 DirectProductGog
  24 [33X[0;0YCat-1-groups[133X
    24.1 [33X[0;0Y [133X
      24.1-1 AutomorphismGroupAsCatOneGroup
      24.1-2 HomotopyGroup
      24.1-3 HomotopyModule
      24.1-4 QuasiIsomorph
      24.1-5 ModuleAsCatOneGroup
      24.1-6 MooreComplex
      24.1-7 NormalSubgroupAsCatOneGroup
      24.1-8 XmodToHAP
  25 [33X[0;0YSimplicial groups[133X
    25.1 [33X[0;0Y [133X
      25.1-1 NerveOfCatOneGroup
      25.1-2 EilenbergMacLaneSimplicialGroup
      25.1-3 EilenbergMacLaneSimplicialGroupMap
      25.1-4 MooreComplex
      25.1-5 ChainComplexOfSimplicialGroup
      25.1-6 SimplicialGroupMap
      25.1-7 HomotopyGroup
      25.1-8 Representation of elements in the bar resolution
      25.1-9 BarResolutionBoundary
      25.1-10 BarResolutionHomotopy
      25.1-11 Representation of elements in the bar complex
      25.1-12 BarComplexBoundary
      25.1-13 BarResolutionEquivalence
      25.1-14 BarComplexEquivalence
      25.1-15 Representation of elements in the bar cocomplex
      25.1-16 BarCocomplexCoboundary
  26 [33X[0;0YCoxeter diagrams and graphs of groups[133X
    26.1 [33X[0;0Y [133X
      26.1-1 CoxeterDiagramComponents
      26.1-2 CoxeterDiagramDegree
      26.1-3 CoxeterDiagramDisplay
      26.1-4 CoxeterDiagramFpArtinGroup
      26.1-5 CoxeterDiagramFpCoxeterGroup
      26.1-6 CoxeterDiagramIsSpherical
      26.1-7 CoxeterDiagramMatrix
      26.1-8 CoxeterSubDiagram
      26.1-9 CoxeterDiagramVertices
      26.1-10 EvenSubgroup
      26.1-11 GraphOfGroupsDisplay
      26.1-12 GraphOfResolutions
      26.1-13 GraphOfGroups
      26.1-14 GraphOfResolutionsDisplay
      26.1-15 GraphOfGroupsTest
      26.1-16 TreeOfGroupsToContractibleGcomplex
      26.1-17 TreeOfResolutionsToContractibleGcomplex
  27 [33X[0;0YTorsion Subcomplexes[133X
    27.1 [33X[0;0Y [133X
      27.1-1 RigidFacetsSubdivision
      27.1-2 IsPNormal
      27.1-3 TorsionSubcomplex
      27.1-4 DisplayAvailableCellComplexes
      27.1-5 VisualizeTorsionSkeleton
      27.1-6 ReduceTorsionSubcomplex
      27.1-7 EquivariantEulerCharacteristic
      27.1-8 CountingCellsOfACellComplex
      27.1-9 CountingControlledSubdividedCells
      27.1-10 CountingBaryCentricSubdividedCells
      27.1-11 EquivariantSpectralSequencePage
      27.1-12 ExportHapCellcomplexToDisk
  28 [33X[0;0YSimplicial Complexes[133X
    28.1 [33X[0;0Y [133X
      28.1-1 Homology
      28.1-2 RipsHomology
      28.1-3 Bettinumbers
      28.1-4 ChainComplex
      28.1-5 CechComplexOfPureCubicalComplex
      28.1-6 PureComplexToSimplicialComplex
      28.1-7 RipsChainComplex
      28.1-8 VectorsToSymmetricMatrix
      28.1-9 EulerCharacteristic
      28.1-10 MaximalSimplicesToSimplicialComplex
      28.1-11 SkeletonOfSimplicialComplex
      28.1-12 GraphOfSimplicialComplex
      28.1-13 ContractibleSubcomplexOfSimplicialComplex
      28.1-14 PathComponentsOfSimplicialComplex
      28.1-15 QuillenComplex
      28.1-16 SymmetricMatrixToIncidenceMatrix
      28.1-17 IncidenceMatrixToGraph
      28.1-18 CayleyGraphOfGroup
      28.1-19 PathComponentsOfGraph
      28.1-20 ContractGraph
      28.1-21 GraphDisplay
      28.1-22 SimplicialMap
      28.1-23 ChainMapOfSimplicialMap
      28.1-24 SimplicialNerveOfGraph
  29 [33X[0;0YCubical Complexes[133X
    29.1 [33X[0;0Y [133X
      29.1-1 ArrayToPureCubicalComplex
      29.1-2 PureCubicalComplex
      29.1-3 FramedPureCubicalComplex
      29.1-4 RandomCubeOfPureCubicalComplex
      29.1-5 PureCubicalComplexIntersection
      29.1-6 PureCubicalComplexUnion
      29.1-7 PureCubicalComplexDifference
      29.1-8 ReadImageAsPureCubicalComplex
      29.1-9 ReadLinkImageAsPureCubicalComplex
      29.1-10 ReadImageSequenceAsPureCubicalComplex
      29.1-11 Size
      29.1-12 Dimension
      29.1-13 WritePureCubicalComplexAsImage
      29.1-14 ViewPureCubicalComplex
      29.1-15 Homology
      29.1-16 Bettinumbers
      29.1-17 DirectProductOfPureCubicalComplexes
      29.1-18 SuspensionOfPureCubicalComplex
      29.1-19 EulerCharacteristic
      29.1-20 PathComponentOfPureCubicalComplex
      29.1-21 ChainComplex
      29.1-22 ChainComplexOfPair
      29.1-23 ExcisedPureCubicalPair
      29.1-24 ChainInclusionOfPureCubicalPair
      29.1-25 ChainMapOfPureCubicalPairs
      29.1-26 ContractPureCubicalComplex
      29.1-27 ContractedComplex
      29.1-28 ZigZagContractedPureCubicalComplex
      29.1-29 ContractCubicalComplex
      29.1-30 DVFReducedCubicalComplex
      29.1-31 SkeletonOfCubicalComplex
      29.1-32 ContractibleSubomplexOfPureCubicalComplex
      29.1-33 AcyclicSubomplexOfPureCubicalComplex
      29.1-34 HomotopyEquivalentMaximalPureCubicalSubcomplex
      29.1-35 HomotopyEquivalentMinimalPureCubicalSubcomplex
      29.1-36 BoundaryOfPureCubicalComplex
      29.1-37 SingularitiesOfPureCubicalComplex
      29.1-38 ThickenedPureCubicalComplex
      29.1-39 CropPureCubicalComplex
      29.1-40 BoundingPureCubicalComplex
      29.1-41 MorseFiltration
      29.1-42 ComplementOfPureCubicalComplex
      29.1-43 PureCubicalComplexToTextFile
      29.1-44 ThickeningFiltration
      29.1-45 Dendrogram
      29.1-46 DendrogramDisplay
      29.1-47 DendrogramToPersistenceMat
      29.1-48 ReadImageAsFilteredPureCubicalComplex
      29.1-49 ComplementOfFilteredPureCubicalComplex
      29.1-50 PersistentHomologyOfFilteredPureCubicalComplex
  30 [33X[0;0YRegular CW-Complexes[133X
    30.1 [33X[0;0Y [133X
      30.1-1 SimplicialComplexToRegularCWComplex
      30.1-2 CubicalComplexToRegularCWComplex
      30.1-3 CriticalCellsOfRegularCWComplex
      30.1-4 ChainComplex
      30.1-5 ChainComplexOfRegularCWComplex
      30.1-6 FundamentalGroup
  31 [33X[0;0YKnots and Links[133X
    31.1 [33X[0;0Y [133X
      31.1-1 PureCubicalKnot
      31.1-2 ViewPureCubicalKnot
      31.1-3 KnotSum
      31.1-4 KnotGroup
      31.1-5 AlexanderMatrix
      31.1-6 AlexanderPolynomial
      31.1-7 ProjectionOfPureCubicalComplex
      31.1-8 ReadPDBfileAsPureCubicalComplex
  32 [33X[0;0YKnots and Quandles[133X
    32.1 [33X[0;0Y [133X
      32.1-1 PresentationKnotQuandle
      32.1-2 PD2GC
      32.1-3 PlanarDiagramKnot
      32.1-4 GaussCodeKnot
      32.1-5 PresentationKnotQuandleKnot
      32.1-6 NumberOfHomomorphisms
      32.1-7 PartitionedNumberOfHomomorphisms
      32.1-8 ConjugationQuandle
      32.1-9 FirstQuandleAxiomIsSatisfied
      32.1-10 IsQuandle
      32.1-11 Quandles
      32.1-12 Quandle
      32.1-13 IdQuandle
      32.1-14 IsLatin
      32.1-15 IsConnectedQuandle
      32.1-16 ConnectedQuandles
      32.1-17 ConnectedQuandle
      32.1-18 IdConnectedQuandle
      32.1-19 IsQuandleEnvelope
      32.1-20 QuandleQuandleEnvelope
      32.1-21 KnotInvariantCedric
      32.1-22 RightMultiplicationGroupAsPerm
      32.1-23 RightMultiplicationGroup
      32.1-24 AutomorphismGroupQuandleAsPerm
      32.1-25 AutomorphismGroupQuandle
  33 [33X[0;0YFinite metric spaces and their filtered complexes[133X
    33.1 [33X[0;0Y [133X
      33.1-1 CayleyMetric
      33.1-2 HammingMetric
      33.1-3 KendallMetric
      33.1-4 EuclideanSquaredMetric
      33.1-5 EuclideanApproximatedMetric
      33.1-6 ManhattanMetric
      33.1-7 VectorsToSymmetricMatrix
      33.1-8 SymmetricMatDisplay
      33.1-9 SymmetricMatrixToFilteredGraph
      33.1-10 PermGroupToFilteredGraph
  34 [33X[0;0YCommutative diagrams and abstract categories[133X
    34.1 [33X[0;0Y [133X
      34.1-1 HomomorphismChainToCommutativeDiagram
      34.1-2 NormalSeriesToQuotientDiagram
      34.1-3 NerveOfCommutativeDiagram
      34.1-4 GroupHomologyOfCommutativeDiagram
      34.1-5 PersistentHomologyOfCommutativeDiagramOfPGroups
    34.2 [33X[0;0Y [133X
      34.2-1 CategoricalEnrichment
      34.2-2 IdentityArrow
      34.2-3 InitialArrow
      34.2-4 TerminalArrow
      34.2-5 HasInitialObject
      34.2-6 HasTerminalObject
      34.2-7 Source
      34.2-8 Target
      34.2-9 CategoryName
      34.2-10 CompositionEqualityAdditionMinus
      34.2-11 Object
      34.2-12 Mapping
      34.2-13 IsCategoryObject
      34.2-14 IsCategoryArrow
  35 [33X[0;0YArrays and Pseudo lists[133X
    35.1 [33X[0;0Y [133X
      35.1-1 Array
      35.1-2 PermuteArray
      35.1-3 ArrayDimension
      35.1-4 ArrayDimensions
      35.1-5 ArraySum
      35.1-6 ArrayValue
      35.1-7 ArrayValueFunctions
      35.1-8 ArrayAssign
      35.1-9 ArrayAssignFunctions
      35.1-10 ArrayIterate
      35.1-11 BinaryArrayToTextFile
      35.1-12 FrameArray
      35.1-13 UnframeArray
      35.1-14 Add
      35.1-15 Append
      35.1-16 ListToPseudoList
  36 [33X[0;0YParallel Computation - Core Functions[133X
    36.1 [33X[0;0Y [133X
      36.1-1 ChildProcess
      36.1-2 ChildClose
      36.1-3 ChildCommand
      36.1-4 NextAvailableChild
      36.1-5 IsAvailableChild
      36.1-6 ChildPut
      36.1-7 ChildGet
      36.1-8 HAPPrintTo
      36.1-9 HAPRead
  37 [33X[0;0YParallel Computation - Extra Functions[133X
    37.1 [33X[0;0Y [133X
      37.1-1 ChildFunction
      37.1-2 ChildRead
      37.1-3 ChildReadEval
      37.1-4 ParallelList
  38 [33X[0;0YSome functions for accessing basic data[133X
    38.1 [33X[0;0Y [133X
      38.1-1 BoundaryMap
      38.1-2 BoundaryMatrix
      38.1-3 Dimension
      38.1-4 EvaluateProperty
      38.1-5 GroupOfResolution
      38.1-6 Length
      38.1-7 Map
      38.1-8 Source
      38.1-9 Target
  39 [33X[0;0YMiscellaneous[133X
    39.1 [33X[0;0Y [133X
      39.1-1 SL2Z
      39.1-2 BigStepLCS
      39.1-3 Classify
      39.1-4 RefineClassification
      39.1-5 Compose
      39.1-6 HAPcopyright
      39.1-7 IsLieAlgebraHomomorphism
      39.1-8 IsSuperperfect
      39.1-9 MakeHAPManual
      39.1-10 PermToMatrixGroup
      39.1-11 SolutionsMatDestructive
      39.1-12 LinearHomomorphismsPersistenceMat
      39.1-13 NormalSeriesToQuotientHomomorphisms
      39.1-14 TestHap
  40 [33X[0;0YHAP variables that are not yet documented[133X
    40.1 [33X[0;0Y [133X
  
  
  [32X
