Regina Calculation Engine

Represents a finitely generated abelian group given by a chain complex. More...
#include <algebra/nmarkedabeliangroup.h>
Public Member Functions  
NMarkedAbelianGroup (const NMatrixInt &M, const NMatrixInt &N)  
Creates a marked abelian group from a chain complex. More...  
NMarkedAbelianGroup (const NMatrixInt &M, const NMatrixInt &N, const NLargeInteger &pcoeff)  
Creates a marked abelian group from a chain complex with coefficients in Z_p. More...  
NMarkedAbelianGroup (unsigned long rk, const NLargeInteger &p)  
Creates a free Z_pmodule of a given rank using the direct sum of the standard chain complex 0 –> Z –p–> Z –> 0 . More...  
NMarkedAbelianGroup (const NMarkedAbelianGroup &cloneMe)  
Creates a clone of the given group. More...  
bool  isChainComplex () const 
Determines whether or not the defining maps for this group actually give a chain complex. More...  
~NMarkedAbelianGroup ()  
Destroys the group. More...  
unsigned long  getRank () const 
Returns the rank of the group. More...  
unsigned long  getTorsionRank (const NLargeInteger °ree) const 
Returns the rank in the group of the torsion term of given degree. More...  
unsigned long  getTorsionRank (unsigned long degree) const 
Returns the rank in the group of the torsion term of given degree. More...  
unsigned long  getNumberOfInvariantFactors () const 
Returns the number of invariant factors that describe the torsion elements of this group. More...  
unsigned long  minNumberOfGenerators () const 
Returns the minimum number of generators for the group. More...  
const NLargeInteger &  getInvariantFactor (unsigned long index) const 
Returns the given invariant factor describing the torsion elements of this group. More...  
bool  isTrivial () const 
Determines whether this is the trivial (zero) group. More...  
bool  operator== (const NMarkedAbelianGroup &other) const 
Determines whether this and the given abelian group are isomorphic. More...  
bool  isIsomorphicTo (const NMarkedAbelianGroup &other) const 
Determines whether this and the given abelian group are isomorphic. More...  
bool  equalTo (const NMarkedAbelianGroup &other) const 
Determines whether or not the two NMarkedAbelianGroups are identical, which means they have exactly the same presentation matrices. More...  
void  writeTextShort (std::ostream &out) const 
The text representation will be of the form 3 Z + 4 Z_2 + Z_120 . More...  
std::vector< NLargeInteger >  getFreeRep (unsigned long index) const 
Returns the requested free generator in the original chain complex defining the group. More...  
std::vector< NLargeInteger >  getTorsionRep (unsigned long index) const 
Returns the requested generator of the torsion subgroup but represented in the original chain complex defining the group. More...  
std::vector< NLargeInteger >  ccRep (const std::vector< NLargeInteger > &SNFRep) const 
A combination of getFreeRep and getTorsion rep, this routine takes a vector which represents an element in the group in the SNF coordinates and returns a corresponding vector in the original chain complex. More...  
std::vector< NLargeInteger >  ccRep (unsigned long SNFRep) const 
Same as ccRep(const std::vector<NLargeInteger>&), but we assume you only want the chain complex representation of a standard basis vector from SNF coordinates. More...  
std::vector< NLargeInteger >  cycleProjection (const std::vector< NLargeInteger > &ccelt) const 
Projects an element of the chain complex to the subspace of cycles. More...  
std::vector< NLargeInteger >  cycleProjection (unsigned long ccindx) const 
Projects an element of the chain complex to the subspace of cycles. More...  
bool  isCycle (const std::vector< NLargeInteger > &input) const 
Given a vector, determines if it represents a cycle in the chain complex. More...  
std::vector< NLargeInteger >  boundaryMap (const std::vector< NLargeInteger > &CCrep) const 
Computes the differential of the given vector in the chain complex whose kernel is the cycles. More...  
bool  isBoundary (const std::vector< NLargeInteger > &input) const 
Given a vector, determines if it represents a boundary in the chain complex. More...  
std::vector< NLargeInteger >  writeAsBoundary (const std::vector< NLargeInteger > &input) const 
Expresses the given vector as a boundary in the chain complex (if the vector is indeed a boundary at all). More...  
unsigned long  getRankCC () const 
Returns the rank of the chain complex supporting the homology computation. More...  
std::vector< NLargeInteger >  snfRep (const std::vector< NLargeInteger > &v) const 
Expresses the given vector as a combination of free and torsion generators. More...  
std::vector< NLargeInteger >  getSNFIsoRep (const std::vector< NLargeInteger > &v) const 
A deprecated alternative to snfRep(). More...  
unsigned long  minNumberCycleGens () const 
Returns the number of generators of ker(M), where M is one of the defining matrices of the chain complex. More...  
std::vector< NLargeInteger >  cycleGen (unsigned long i) const 
Returns the ith generator of the cycles, i.e., the kernel of M in the chain complex. More...  
const NMatrixInt &  getMRB () const 
Returns a changeofbasis matrix for the Smith normal form of M. More...  
const NMatrixInt &  getMRBi () const 
Returns an inverse changeofbasis matrix for the Smith normal form of M. More...  
const NMatrixInt &  getMCB () const 
Returns a changeofbasis matrix for the Smith normal form of M. More...  
const NMatrixInt &  getMCBi () const 
Returns an inverse changeofbasis matrix for the Smith normal form of M. More...  
const NMatrixInt &  getNRB () const 
Returns a changeofbasis matrix for the Smith normal form of the internal presentation matrix. More...  
const NMatrixInt &  getNRBi () const 
Returns an inverse changeofbasis matrix for the Smith normal form of the internal presentation matrix. More...  
const NMatrixInt &  getNCB () const 
Returns a changeofbasis matrix for the Smith normal form of the internal presentation matrix. More...  
const NMatrixInt &  getNCBi () const 
Returns an inverse changeofbasis matrix for the Smith normal form of the internal presentation matrix. More...  
unsigned long  getRankM () const 
Returns the rank of the defining matrix M. More...  
unsigned long  getFreeLoc () const 
Returns the index of the first free generator in the Smith normal form of the internal presentation matrix. More...  
unsigned long  getTorsionLoc () const 
Returns the index of the first torsion generator in the Smith normal form of the internal presentation matrix. More...  
const NMatrixInt &  getM () const 
Returns the `right' matrix used in defining the chain complex. More...  
const NMatrixInt &  getN () const 
Returns the `left' matrix used in defining the chain complex. More...  
const NLargeInteger &  coefficients () const 
Returns the coefficients used for the computation of homology. More...  
std::auto_ptr < NMarkedAbelianGroup >  torsionSubgroup () const 
Returns an NMarkedAbelianGroup representing the torsion subgroup of this group. More...  
std::auto_ptr < NHomMarkedAbelianGroup >  torsionInclusion () const 
Returns an NHomMarkedAbelianGroup representing the inclusion of the torsion subgroup into this group. More...  
Input and Output  
virtual void  writeTextLong (std::ostream &out) const 
Writes this object in long text format to the given output stream. More...  
std::string  str () const 
Returns the output from writeTextShort() as a string. More...  
std::string  toString () const 
A deprecated alias for str(), which returns the output from writeTextShort() as a string. More...  
std::string  detail () const 
Returns the output from writeTextLong() as a string. More...  
std::string  toStringLong () const 
A deprecated alias for detail(), which returns the output from writeTextLong() as a string. More...  
Friends  
class  NHomMarkedAbelianGroup 
Represents a finitely generated abelian group given by a chain complex.
This class is initialized with a chain complex. The chain complex is given in terms of two integer matrices M and N such that M*N=0. The abelian group is the kernel of M mod the image of N.
In other words, we are computing the homology of the chain complex Z^a –N–> Z^b –M–> Z^c
where a=N.columns(), M.columns()=b=N.rows(), and c=M.rows(). An additional constructor allows one to take the homology with coefficients in an arbitrary cyclic group.
This class allows one to retrieve the invariant factors, the rank, and the corresponding vectors in the kernel of M. Moreover, given a vector in the kernel of M, it decribes the homology class of the vector (the free part, and its position in the invariant factors).
The purpose of this class is to allow one to not only represent homology groups, but it gives coordinates on the group allowing for the construction of homomorphisms, and keeping track of subgroups.
Some routines in this class refer to the internal presentation matrix. This is a proper presentation matrix for the abelian group, and is created by constructing the product getMRBi() * N, and then removing the first getRankM() rows.
Optimise (longterm): Look at using sparse matrices for storage of SNF and the like.
Testsuite additions: isBoundary(), boundaryMap(), writeAsBdry(), cycleGen().
regina::NMarkedAbelianGroup::NMarkedAbelianGroup  (  const NMatrixInt &  M, 
const NMatrixInt &  N  
) 
Creates a marked abelian group from a chain complex.
This constructor assumes you're interested in homology with integer coefficents of the chain complex. Creates a marked abelian group given by the quotient of the kernel of M modulo the image of N.
See the class notes for further details.
M  the `right' matrix in the chain complex; that is, the matrix that one takes the kernel of when computing homology. 
N  the `left' matrix in the chain complex; that is, the matrix that one takes the image of when computing homology. 
regina::NMarkedAbelianGroup::NMarkedAbelianGroup  (  const NMatrixInt &  M, 
const NMatrixInt &  N,  
const NLargeInteger &  pcoeff  
) 
Creates a marked abelian group from a chain complex with coefficients in Z_p.
M  the `right' matrix in the chain complex; that is, the matrix that one takes the kernel of when computing homology. 
N  the `left' matrix in the chain complex; that is, the matrix that one takes the image of when computing homology. 
pcoeff  specifies the coefficient ring, Z_pcoeff. We require pcoeff >= 0. If you know beforehand that pcoeff=0, it's more efficient to use the previous constructor. 
regina::NMarkedAbelianGroup::NMarkedAbelianGroup  (  unsigned long  rk, 
const NLargeInteger &  p  
) 
Creates a free Z_pmodule of a given rank using the direct sum of the standard chain complex 0 –> Z –p–> Z –> 0
.
So this group is isomorphic to n Z_p
. Moreover, if constructed using the previous constructor, M would be zero and N would be diagonal and square with p down the diagonal.
rk  the rank of the group as a Z_pmodule. That is, if the group is n Z_p , then rk should be n. 
p  describes the type of ring that we use to talk about the "free" module. 

inline 
Creates a clone of the given group.
cloneMe  the group to clone. 

inline 
Destroys the group.
std::vector<NLargeInteger> regina::NMarkedAbelianGroup::boundaryMap  (  const std::vector< NLargeInteger > &  CCrep  )  const 
Computes the differential of the given vector in the chain complex whose kernel is the cycles.
In other words, this routine returns M*CCrep
.
CCrep  a vector whose length is M.columns(), where M is one of the matrices that defines the chain complex (see the class notes for details). 
std::vector<NLargeInteger> regina::NMarkedAbelianGroup::ccRep  (  const std::vector< NLargeInteger > &  SNFRep  )  const 
A combination of getFreeRep and getTorsion rep, this routine takes a vector which represents an element in the group in the SNF coordinates and returns a corresponding vector in the original chain complex.
This routine is the inverse to snfRep() described below.
SNFRep  a vector of size the number of generators of the group, i.e., it must be valid in the SNF coordinates. If not, an empty vector is returned. 
std::vector<NLargeInteger> regina::NMarkedAbelianGroup::ccRep  (  unsigned long  SNFRep  )  const 
Same as ccRep(const std::vector<NLargeInteger>&), but we assume you only want the chain complex representation of a standard basis vector from SNF coordinates.
SNFRep  specifies which standard basis vector from SNF coordinates; this must be between 0 and minNumberOfGenerators()1 inclusive. 

inline 
Returns the coefficients used for the computation of homology.
That is, this routine returns the integer p where we use coefficients in Z_p. If we use coefficients in the integers Z, then this routine returns 0.
std::vector<NLargeInteger> regina::NMarkedAbelianGroup::cycleGen  (  unsigned long  i  )  const 
Returns the ith generator of the cycles, i.e., the kernel of M in the chain complex.
i  between 0 and minNumCycleGens()1. 
std::vector<NLargeInteger> regina::NMarkedAbelianGroup::cycleProjection  (  const std::vector< NLargeInteger > &  ccelt  )  const 
Projects an element of the chain complex to the subspace of cycles.
Returns an empty vector if the input element does not have dimensions of the chain complex.
ccelt  a vector whose length is M.columns(), where M is one of the matrices that defines the chain complex (see the class notes for details). 
std::vector<NLargeInteger> regina::NMarkedAbelianGroup::cycleProjection  (  unsigned long  ccindx  )  const 
Projects an element of the chain complex to the subspace of cycles.
Returns an empty vector if the input index is out of bounds.
ccindx  the index of the standard basis vector in chain complex coordinates. 

inherited 
Returns the output from writeTextLong() as a string.

inline 
Determines whether or not the two NMarkedAbelianGroups are identical, which means they have exactly the same presentation matrices.
This is useful for determining if two NHomMarkedAbelianGroups are composable. See isIsomorphicTo() if all you care about is the isomorphism relation among groups defined by presentation matrices.
other  the NMarkedAbelianGroup with which this should be compared. 
true
if and only if the two groups have identical chaincomplex definitions.

inline 
Returns the index of the first free generator in the Smith normal form of the internal presentation matrix.
See the class overview for details.
std::vector<NLargeInteger> regina::NMarkedAbelianGroup::getFreeRep  (  unsigned long  index  )  const 
Returns the requested free generator in the original chain complex defining the group.
As described in the class overview, this marked abelian group is defined by matrices M and N where M*N = 0. If M is an m by l matrix and N is an l by n matrix, then this routine returns the (index)th free generator of ker(M)/img(N) in Z^l.
index  specifies which free generator to look up; this must be between 0 and getRank()1 inclusive. 

inline 
Returns the given invariant factor describing the torsion elements of this group.
See the NMarkedAbelianGroup class notes for further details.
If the invariant factors are d0d1...dn, this routine will return di where i is the value of parameter index.
index  the index of the invariant factor to return; this must be between 0 and getNumberOfInvariantFactors()1 inclusive. 

inline 
Returns the `right' matrix used in defining the chain complex.
Our group was defined as the kernel of M mod the image of N. This is the matrix M.
This is a copy of the matrix M that was originally passed to the class constructor. See the class overview for further details on matrices M and N and their roles in defining the chain complex.

inline 
Returns a changeofbasis matrix for the Smith normal form of M.
This is one of several routines that returns information on how we determine the isomorphismclass of this group.
Recall from the class overview that this marked abelian group is defined by matrices M and N, where M*N = 0.

inline 
Returns an inverse changeofbasis matrix for the Smith normal form of M.
This is one of several routines that returns information on how we determine the isomorphismclass of this group.
Recall from the class overview that this marked abelian group is defined by matrices M and N, where M*N = 0.

inline 
Returns a changeofbasis matrix for the Smith normal form of M.
This is one of several routines that returns information on how we determine the isomorphismclass of this group.
Recall from the class overview that this marked abelian group is defined by matrices M and N, where M*N = 0.

inline 
Returns an inverse changeofbasis matrix for the Smith normal form of M.
This is one of several routines that returns information on how we determine the isomorphismclass of this group.
Recall from the class overview that this marked abelian group is defined by matrices M and N, where M*N = 0.

inline 
Returns the `left' matrix used in defining the chain complex.
Our group was defined as the kernel of M mod the image of N. This is the matrix N.
This is a copy of the matrix N that was originally passed to the class constructor. See the class overview for further details on matrices M and N and their roles in defining the chain complex.

inline 
Returns a changeofbasis matrix for the Smith normal form of the internal presentation matrix.
This is one of several routines that returns information on how we determine the isomorphismclass of this group.
For details on the internal presentation matrix, see the class overview. If P is the internal presentation matrix, then:

inline 
Returns an inverse changeofbasis matrix for the Smith normal form of the internal presentation matrix.
This is one of several routines that returns information on how we determine the isomorphismclass of this group.
For details on the internal presentation matrix, see the class overview. If P is the internal presentation matrix, then:

inline 
Returns a changeofbasis matrix for the Smith normal form of the internal presentation matrix.
This is one of several routines that returns information on how we determine the isomorphismclass of this group.
For details on the internal presentation matrix, see the class overview. If P is the internal presentation matrix, then:

inline 
Returns an inverse changeofbasis matrix for the Smith normal form of the internal presentation matrix.
This is one of several routines that returns information on how we determine the isomorphismclass of this group.
For details on the internal presentation matrix, see the class overview. If P is the internal presentation matrix, then:

inline 
Returns the number of invariant factors that describe the torsion elements of this group.
This is the minimal number of torsion generators. See the NMarkedAbelianGroup class notes for further details.

inline 
Returns the rank of the group.
This is the number of included copies of Z.

inline 
Returns the rank of the chain complex supporting the homology computation.
In the description of this class, this is also given by M.columns() and N.rows() from the constructor that takes as input two matrices, M and N.

inline 
Returns the rank of the defining matrix M.
The matrix M is the `right' matrix used in defining the chain complex. See the class overview for further details.

inline 
A deprecated alternative to snfRep().
v  a vector of length M.columns(). 

inline 
Returns the index of the first torsion generator in the Smith normal form of the internal presentation matrix.
See the class overview for details.
unsigned long regina::NMarkedAbelianGroup::getTorsionRank  (  const NLargeInteger &  degree  )  const 
Returns the rank in the group of the torsion term of given degree.
If the given degree is d, this routine will return the largest m for which mZ_d is a subgroup of this group.
For instance, if this group is Z_6+Z_12, the torsion term of degree 2 has rank 2 (one occurrence in Z_6 and one in Z_12), and the torsion term of degree 4 has rank 1 (one occurrence in Z_12).
degree  the degree of the torsion term to query. 

inline 
Returns the rank in the group of the torsion term of given degree.
If the given degree is d, this routine will return the largest m for which mZ_d is a subgroup of this group.
For instance, if this group is Z_6+Z_12, the torsion term of degree 2 has rank 2 (one occurrence in Z_6 and one in Z_12), and the torsion term of degree 4 has rank 1 (one occurrence in Z_12).
degree  the degree of the torsion term to query. 
std::vector<NLargeInteger> regina::NMarkedAbelianGroup::getTorsionRep  (  unsigned long  index  )  const 
Returns the requested generator of the torsion subgroup but represented in the original chain complex defining the group.
As described in the class overview, this marked abelian group is defined by matrices M and N where M*N = 0. If M is an m by l matrix and N is an l by n matrix, then this routine returns the (index)th torsion generator of ker(M)/img(N) in Z^l.
index  specifies which generator in the torsion subgroup; this must be at least 0 and strictly less than the number of nontrivial invariant factors. If not, you receive an empty vector. 
bool regina::NMarkedAbelianGroup::isBoundary  (  const std::vector< NLargeInteger > &  input  )  const 
Given a vector, determines if it represents a boundary in the chain complex.
input  a vector whose length is M.columns(), where M is one of the matrices that defines the chain complex (see the class notes for details). 
true
if and only if the given vector represents a boundary. bool regina::NMarkedAbelianGroup::isChainComplex  (  )  const 
Determines whether or not the defining maps for this group actually give a chain complex.
This is helpful for debugging.
Specifically, this routine returns true
if and only if M*N = 0 where M and N are the definining matrices.
true
if and only if M*N = 0. bool regina::NMarkedAbelianGroup::isCycle  (  const std::vector< NLargeInteger > &  input  )  const 
Given a vector, determines if it represents a cycle in the chain complex.
input  an input vector in chain complex coordinates. 
true
if and only if the given vector represents a cycle.

inline 
Determines whether this and the given abelian group are isomorphic.
other  the group with which this should be compared. 
true
if and only if the two groups are isomorphic.

inline 
Determines whether this is the trivial (zero) group.
true
if and only if this is the trivial group.

inline 
Returns the number of generators of ker(M), where M is one of the defining matrices of the chain complex.

inline 
Returns the minimum number of generators for the group.

inline 
Determines whether this and the given abelian group are isomorphic.
other  the group with which this should be compared. 
true
if and only if the two groups are isomorphic. std::vector<NLargeInteger> regina::NMarkedAbelianGroup::snfRep  (  const std::vector< NLargeInteger > &  v  )  const 
Expresses the given vector as a combination of free and torsion generators.
This answer is coordinate dependant, meaning the answer may change depending on how the Smith Normal Form is computed.
Recall that this marked abelian was defined by matrices M and N with M*N=0; suppose that M is an m by l matrix and N is an l by n matrix. This abelian group is then the quotient ker(M)/img(N) in Z^l.
When it is constructed, this group is computed to be isomorphic to some Z_{d0} + ... + Z_{dk} + Z^d, where:
This routine takes a single argument v, which must be a vector in Z^l.
If v belongs to ker(M), this routine describes how it projects onto the group ker(M)/img(N). Specifically, it returns a vector of length d + k, where:
In other words, suppose v belongs to ker(M) and snfRep(v) returns the vector (b1, ..., bk, a1, ..., ad). Suppose furthermore that the free generators returned by getFreeRep(0..(d1)) are f1, ..., fd respectively, and that the torsion generators returned by getTorsionRep(0..(k1)) are t1, ..., tk respectively. Then v = b1.t1 + ... + bk.tk + a1.f1 + ... + ad.fd modulo img(N).
If v does not belong to ker(M), this routine simply returns the empty vector.
v  a vector of length M.columns(). M.columns() is also getRankCC(). 

inherited 
Returns the output from writeTextShort() as a string.
__str__()
function.std::auto_ptr<NHomMarkedAbelianGroup> regina::NMarkedAbelianGroup::torsionInclusion  (  )  const 
Returns an NHomMarkedAbelianGroup representing the inclusion of the torsion subgroup into this group.
std::auto_ptr<NMarkedAbelianGroup> regina::NMarkedAbelianGroup::torsionSubgroup  (  )  const 
Returns an NMarkedAbelianGroup representing the torsion subgroup of this group.

inlineinherited 
A deprecated alias for str(), which returns the output from writeTextShort() as a string.

inlineinherited 
A deprecated alias for detail(), which returns the output from writeTextLong() as a string.
std::vector<NLargeInteger> regina::NMarkedAbelianGroup::writeAsBoundary  (  const std::vector< NLargeInteger > &  input  )  const 
Expresses the given vector as a boundary in the chain complex (if the vector is indeed a boundary at all).
This routine uses chain complex coordinates for both the input and the return value.
Nv=input
.

inlinevirtualinherited 
Writes this object in long text format to the given output stream.
The output should provide the user with all the information they could want. The output should be humanreadable, should not contain extremely long lines (so users can read the output in a terminal), and should end with a final newline.
The default implementation of this routine merely calls writeTextShort() and adds a newline.
out  the output stream to which to write. 
Reimplemented in regina::NSnapPeaTriangulation, regina::NGroupPresentation, regina::NHomMarkedAbelianGroup, regina::NNormalSurfaceList, regina::NTetrahedron, regina::NSatRegion, regina::NVertex, regina::NEdge, regina::Dim2Triangle, regina::NTriangle, regina::NLayeredSolidTorus, regina::NHomGroupPresentation, regina::NGenericIsomorphism< dim >, regina::NGenericIsomorphism< 2 >, regina::NGenericIsomorphism< 3 >, regina::NTriangulation, regina::NComponent, regina::NTxICore, regina::NTriSolidTorus, regina::NAngleStructureList, regina::Dim2Edge, regina::NBoundaryComponent, regina::NLayeredChain, regina::Dim2Vertex, regina::Dim2Component, regina::NScript, regina::NAugTriSolidTorus, regina::NSpiralSolidTorus, regina::NSurfaceFilterProperties, regina::NLayeredTorusBundle, regina::NManifold, regina::NPlugTriSolidTorus, regina::NMatrixInt, regina::NBlockedSFSTriple, regina::NPluggedTorusBundle, regina::Dim2Triangulation, regina::NSurfaceSubset, regina::NLayeredLensSpace, regina::NLayeredLoop, regina::NFileInfo, regina::NBlockedSFSLoop, regina::NSnappedBall, regina::NBlockedSFSPair, regina::Dim2BoundaryComponent, regina::NTrivialTri, regina::NL31Pillow, regina::NLayeredChainPair, regina::NText, regina::NSurfaceFilterCombination, and regina::NBlockedSFS.

virtual 
The text representation will be of the form 3 Z + 4 Z_2 + Z_120
.
The torsion elements will be written in terms of the invariant factors of the group, as described in the NMarkedAbelianGroup notes.
out  the stream to write to. 
Implements regina::ShareableObject.