Simplicial Homology
- Introduction
- Simplicial Complexes
SimplicialComplex(f): SeqEnum[SetEnum] → SmpCpxExample: constructSimplicialComplex(G): Grph → SmpCpxFlagComplex(G): Grph → SmpCpxCliqueComplex(G): Grph → SmpCpxDimension(X): SmpCpx → RngIntEltExample: dimensionFaces(X, d): SmpCpx, RngIntElt → SeqEnum[SetEnum]Example: facesFacets(X): SmpCpx → SeqEnum[SetEnum]Example: facetsNormalization(X): SmpCpx → SmpCpxNormalization(~X): SmpCpxShift(X, n): SmpCpx, RngIntElt → SmpCpxShift(~X, n): SmpCpx, RngIntEltExample: Normalize ShiftBoundary(X): SmpCpx → SmpCpxExample: boundaryS + T: SmpCpx, SmpCpx → SmpCpxExample: sumS eq T: SmpCpx, SmpCpx → BoolEltExample: eqProduct(S, T): SmpCpx, SmpCpx → SmpCpxExample: productJoin(S, T): SmpCpx, SmpCpx → SmpCpxS * T: SmpCpx, SmpCpx → SmpCpxExample: joinAddSimplex(X, s): SmpCpx, SetEnum → SmpCpxAddSimplex(~X, s): SmpCpx, SetEnumAddSimplex(X, s): SmpCpx, SeqEnum → SmpCpxAddSimplex(~X, s): SmpCpx, SeqEnumPrune(X, f): SmpCpx, SetEnum → SmpCpxPrune(~X, f): SmpCpx, SetEnumGlue(X, e): SmpCpx, SeqEnum → SmpCpxGlue(~X, e): SmpCpx, SeqEnumExample: Prune GlueBarycentricSubdivision(X): SmpCpx → SmpCpxSkeleton(X, q): SmpCpx, RngIntElt → SmpCpxUnderlyingGraph(X): SmpCpx → GrphUnd, GrphVertSet, GrphEdgeSetCone(X): SmpCpx → SmpCpxSuspension(X): SmpCpx → SmpCpxExample: Cone Suspension- Standard Topological Objects
- Homology Computation
Homology(X): SmpCpx → SeqEnum, SeqEnumHomology(~X): SmpCpxHomology(X, A): SmpCpx, Rng → SeqEnum, SeqEnumHomology(~X, A): SmpCpx, RngExample: homologyHomologyGroup(X, q): SmpCpx, RngIntElt → ModRngHomologyGroup(X, q, A): SmpCpx, RngIntElt, Rng → ModRngBettiNumber(X, q): SmpCpx, RngIntElt → RngIntEltBettiNumber(X, q, A): SmpCpx, RngIntElt, Rng → RngIntEltTorsionCoefficients(X, q): SmpCpx, RngIntElt → SeqEnum[RngElt]TorsionCoefficients(X, q, A): SmpCpx, RngIntElt, Rng → SeqEnum[RngElt]EulerCharacteristic(X): SmpCpx → RngIntEltBoundaryMatrix(X, q, A): SmpCpx, RngIntElt, Rng → MtrxChainComplex(X, A): SmpCpx, Rng → ModCpxExample: chaincomplexHomologyGenerators(X): SmpCpxHomologyGenerators(X, A): SmpCpx, RngHomologyGenerators(H, M, X): SeqEnum, SeqEnum, SmpCpxExample: homologygenerators
