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regina::NSignature Class Reference
[Splitting Surfaces]

Represents a signature of a splitting surface in a closed 3-manifold triangulation. More...

#include <nsignature.h>

Inheritance diagram for regina::NSignature:

regina::ShareableObject regina::boost::noncopyable List of all members.

Public Member Functions

 NSignature (const NSignature &sig)
 Creates a new signature that is a clone of the given signature.

virtual ~NSignature ()
 Destroys this signature.

unsigned getOrder () const
 Returns the order of this signature.

NTriangulationtriangulate () const
 Returns a newly created 3-manifold triangulation corresponding to this splitting surface signature.

void writeCycles (std::ostream &out, const std::string &cycleOpen, const std::string &cycleClose, const std::string &cycleJoin) const
 Writes a string representation of this signature to the given output stream.

virtual void writeTextShort (std::ostream &out) const
 Writes this object in short text format to the given output stream.


Static Public Member Functions

NSignatureparse (const std::string &sig)
 Parses the given signature string.

int cycleCmp (const NSignature &sig1, unsigned cycle1, unsigned start1, int dir1, unsigned *relabel1, const NSignature &sig2, unsigned cycle2, unsigned start2, int dir2, unsigned *relabel2)
 Lexicographically compares the results of transformations upon two given cycles.


Detailed Description

Represents a signature of a splitting surface in a closed 3-manifold triangulation.

A splitting surface is (for these purposes) a compact normal surface consisting of precisely one quad per tetrahedron and no other normal (or almost normal) discs.

A signature of order n is a string consisting of 2n letters arranged into cycles, where n is the number of quads in the splitting surface. From a signature, the corresponding splitting surface and then the entire 3-manifold triangulation can be recreated.

A signature of order n uses the first n letters of the alphabet, each precisely twice. Case is important; the meaning of a letter changes according to whether it appears in upper-case or lower-case.

Each letter represents an individual quadrilateral (the two occurrences of the letter representing the quadrilateral's two sides). Each cycle represents a chain of quadrilaterals joined together in the splitting surface. The case of a letter represents in which direction a quadrilateral is traversed within a cycle.

Cycles are arranged into cycle groups, where a cycle group consists of a series of consecutive cycles all of the same length.

An example of a signature is (abc)(a)(b)(c). This signature is of order 3 and contains two cycle groups, the first being (abc) and the second being (a)(b)(c).

A signature cannot represent a splitting surface with more than 26 quadrilaterals.

For further details on splitting surfaces and their signatures, consult Minimal triangulations and normal surfaces, Burton, PhD thesis, available from the Regina website.


Constructor & Destructor Documentation

regina::NSignature::NSignature const NSignature sig  ) 
 

Creates a new signature that is a clone of the given signature.

Parameters:
sig the signature to clone.

regina::NSignature::~NSignature  )  [inline, virtual]
 

Destroys this signature.


Member Function Documentation

int regina::NSignature::cycleCmp const NSignature sig1,
unsigned  cycle1,
unsigned  start1,
int  dir1,
unsigned *  relabel1,
const NSignature sig2,
unsigned  cycle2,
unsigned  start2,
int  dir2,
unsigned *  relabel2
[static]
 

Lexicographically compares the results of transformations upon two given cycles.

Even if transformations are specified, the underlying signatures will not be changed.

This comparison is not case-sensitive.

Precondition:
The two specified cycles have the same length.
Python:
Not present.
Parameters:
sig1 the signature containing the first cycle to examine.
cycle1 specifies which cycle to examine in signature sig1. This must be less than the total number of cycles in sig1.
start1 allows the first cycle to be transformed by rotation; this parameter is the new starting position of the first cycle. This must be between 0 and sig1.getCycleLength(cycle1)-1 inclusive.
dir1 allows the first cycle to be transformed by reversal; this parameter must be positive to use an unreversed cycle or negative to use a reversed cycle.
relabel1 allows the first cycle to be transformed by relabelling; this parameter must be an array of size at least sig1.getOrder() mapping old labels 0,1,... (representing letters A,B,...) to new labels (which must also be 0,1,..., possibly in a different order). This parameter may be 0 if no relabelling is to be used.
sig2 the signature containing the second cycle to examine.
cycle2 specifies which cycle to examine in signature sig2. This must be less than the total number of cycles in sig2.
start2 allows the second cycle to be transformed by rotation; this parameter is the new starting position of the second cycle. This must be between 0 and sig2.getCycleLength(cycle2)-1 inclusive.
dir2 allows the second cycle to be transformed by reversal; this parameter must be positive to use an unreversed cycle or negative to use a reversed cycle.
relabel2 allows the second cycle to be transformed by relabelling; this parameter must be an array of size at least sig2.getOrder() mapping old labels 0,1,... (representing letters A,B,...) to new labels (which must also be 0,1,..., possibly in a different order). This parameter may be 0 if no relabelling is to be used.
Returns:
-1, 1 or 0 if the transformed first cycle is lexicographically less than, greater than or equal to the transformed second cycle respectively.

unsigned regina::NSignature::getOrder  )  const [inline]
 

Returns the order of this signature.

The order is the number of quads in the corresponding splitting surface.

Returns:
the order of this signature.

NSignature* regina::NSignature::parse const std::string &  sig  )  [static]
 

Parses the given signature string.

Punctuation characters in the given string will be interpreted as separating cycles. All whitespace will be ignored.

Examples of valid signatures are "(ab)(bC)(Ca)" and "AAb-bc-C". See the class notes for further details on what constitutes a valid signature.

Precondition:
The given string contains at least one letter.
Parameters:
sig a string representation of a splitting surface signature.
Returns:
a corresponding newly created signature, or 0 if the given string was invalid.

NTriangulation* regina::NSignature::triangulate  )  const
 

Returns a newly created 3-manifold triangulation corresponding to this splitting surface signature.

Returns:
the corresponding triangulation.

void regina::NSignature::writeCycles std::ostream &  out,
const std::string &  cycleOpen,
const std::string &  cycleClose,
const std::string &  cycleJoin
const
 

Writes a string representation of this signature to the given output stream.

Python:
The parameter out does not exist; standard output will be used.
Parameters:
out the output stream to which to write.
cycleOpen the text to write at the beginning of a cycle (such as "(").
cycleClose the text to write at the end of a cycle (such as ")").
cycleJoin the text to write between two cycles.

void regina::NSignature::writeTextShort std::ostream &  out  )  const [inline, virtual]
 

Writes this object in short text format to the given output stream.

The output should fit on a single line and no newline should be written.

Python:
The parameter out does not exist; standard output will be used.
Parameters:
out the output stream to which to write.

Implements regina::ShareableObject.


The documentation for this class was generated from the following file:
Copyright © 1999-2004, Ben Burton
This software is released under the GNU General Public License.
For further information, or to submit a bug or other problem, please contact Ben Burton (bab@debian.org).