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:

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.
NTriangulation *	triangulate () 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
static NSignature *	parse (const std::string &sig)
	Parses the given signature string.
static 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.
Friends
class	regina::NSigPartialIsomorphism
class	regina::NSigCensus

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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.

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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.

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Member Function Documentation

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.

static 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.

static 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.

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.

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The documentation for this class was generated from the following file:

• nsignature.h