Represents a signature of a splitting surface in a closed 3-manifold triangulation. More...
#include <split/signature.h>
Public Member Functions | |
| Signature (const Signature &sig) | |
| Creates a new signature that is a clone of the given signature. More... | |
| ~Signature () | |
| Destroys this signature. More... | |
| unsigned | order () const |
| Returns the order of this signature. More... | |
| Triangulation< 3 > * | triangulate () const |
| Returns a newly created 3-manifold triangulation corresponding to this splitting surface signature. More... | |
| 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. More... | |
| void | writeTextShort (std::ostream &out) const |
| Writes a short text representation of this object to the given output stream. More... | |
| Signature & | operator= (const Signature &)=delete |
| void | writeTextLong (std::ostream &out) const |
| A default implementation for detailed output. More... | |
| std::string | str () const |
| Returns a short text representation of this object. More... | |
| std::string | utf8 () const |
| Returns a short text representation of this object using unicode characters. More... | |
| std::string | detail () const |
| Returns a detailed text representation of this object. More... | |
Static Public Member Functions | |
| static Signature * | parse (const std::string &sig) |
| Parses the given signature string. More... | |
| static int | cycleCmp (const Signature &sig1, unsigned cycle1, unsigned start1, int dir1, unsigned *relabel1, const Signature &sig2, unsigned cycle2, unsigned start2, int dir2, unsigned *relabel2) |
| Lexicographically compares the results of transformations upon two given cycles. More... | |
Friends | |
| class | regina::SigPartialIsomorphism |
| class | regina::SigCensus |
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.
Member Function Documentation
◆ writeTextLong()
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inlineinherited |
A default implementation for detailed output.
This routine simply calls T::writeTextShort() and appends a final newline.
- Python
- Not present.
- Parameters
-
out the output stream to which to write.
The documentation for this class was generated from the following file:
- split/signature.h