regina::NSatLayering Class Reference
[Standard Triangulations and Subcomplexes]

A degenerate saturated block that is a single tetrahedron wrapped around so that two opposite edges touch. More...

#include <nsatblocktypes.h>

Inheritance diagram for regina::NSatLayering:

regina::NSatBlock regina::ShareableObject regina::boost::noncopyable

List of all members.

Public Member Functions

 NSatLayering (const NSatLayering &cloneMe)
 Constructs a clone of the given block structure.
bool overHorizontal () const
 Does this describe a layering over the horizontal edge of the boundary annulus, or a layering over the diagonal edge?
virtual NSatBlockclone () const
 Returns a newly created clone of this saturated block structure.
virtual void adjustSFS (NSFSpace &sfs, bool reflect) const
 Adjusts the given Seifert fibred space to insert the contents of this saturated block.
virtual void writeTextShort (std::ostream &out) const
 Writes this object in short text format to the given output stream.
virtual void writeAbbr (std::ostream &out, bool tex=false) const
 Writes an abbreviated name or symbol for this block to the given output stream.

Static Public Member Functions

static NSatLayeringisBlockLayering (const NSatAnnulus &annulus, TetList &avoidTets)
 Determines whether the given annulus is a boundary annulus for a block of this type (single layering).

Protected Member Functions

 NSatLayering (bool overHorizontal)
 Constructs a partially initialised block.


Detailed Description

A degenerate saturated block that is a single tetrahedron wrapped around so that two opposite edges touch.

This forms a degenerate one-tetrahedron solid torus that is pinched along a single meridinal curve.

The four faces of this tetrahedron form two boundary annuli, and the tetrahedron is effectively layered onto each boundary annulus. See the NLayering class notes for more discussion on layerings in general.

Although this block is degenerate (the fibres are all pinched together where the opposite edges of the tetrahedron meet), it can be used without problems as long as the entire Seifert fibred space is not formed from degenerate blocks. In other words, using such blocks is fine as long as they eventually meet a real (non-degenerate) block, which will give room for the fibres to separate so that they are no longer pinched together.

The NSatAnnulus class notes describe horizontal and diagonal edges of a saturated annulus. This block may be one of two types, according to how the tetrahedron is layered onto the boundary annuli. Either the tetrahedron can be layered over the horizontal edge of each annulus (with the fibres pinched together between the two diagonal edges), or the tetrahedron can be layered over the diagonal edge of each annulus (with the fibres pinched together between the two horizontal edges).


Constructor & Destructor Documentation

regina::NSatLayering::NSatLayering ( const NSatLayering cloneMe  )  [inline]

Constructs a clone of the given block structure.

Parameters:
cloneMe the block structure to clone.

regina::NSatLayering::NSatLayering ( bool  overHorizontal  )  [inline, protected]

Constructs a partially initialised block.

The boundary annuli will remain uninitialised, and must be initialised before this block can be used.

Parameters:
overHorizontal true if this block describes a layering over the horizontal edge of the boundary annulus, or false if it describes a layering over the diagonal edge.


Member Function Documentation

bool regina::NSatLayering::overHorizontal (  )  const [inline]

Does this describe a layering over the horizontal edge of the boundary annulus, or a layering over the diagonal edge?

See the NSatAnnulus class notes for definitions of horizontal and diagonal in this context.

NSatBlock * regina::NSatLayering::clone (  )  const [inline, virtual]

Returns a newly created clone of this saturated block structure.

A clone of the correct subclass of NSatBlock will be returned. For this reason, each subclass of NSatBlock must implement this routine.

Returns:
a new clone of this block.

Implements regina::NSatBlock.

virtual void regina::NSatLayering::adjustSFS ( NSFSpace sfs,
bool  reflect 
) const [virtual]

Adjusts the given Seifert fibred space to insert the contents of this saturated block.

In particular, the space should be adjusted as though an ordinary solid torus (base orbifold a disc, no twists or exceptional fibres) had been replaced by this block. This description does not make sense for blocks with twisted boundary; the twisted case is discussed below.

If the argument reflect is true, it should be assumed that this saturated block is being reflected before being inserted into the larger Seifert fibred space. That is, any twists or exceptional fibres should be negated before being added.

Regarding the signs of exceptional fibres: Consider a saturated block containing a solid torus whose meridinal curve runs p times horizontally around the boundary in order through annuli 0,1,... and follows the fibres q times from bottom to top (as depicted in the diagram in the NSatBlock class notes). Then this saturated block adds a positive (p, q) fibre to the underlying Seifert fibred space.

If the ring of saturated annuli bounding this block is twisted then the situation becomes more complex. It can be proven that such a block must contain a twisted reflector boundary in the base orbifold (use Z_2 homology with fibre-reversing paths to show that the base orbifold must contain another twisted boundary component, and then recall that real boundaries are not allowed inside blocks).

In this twisted boundary case, it should be assumed that the twisted reflector boundary is already stored in the given Seifert fibred space. This routine should make any further changes that are required (there may well be none). That is, the space should be adjusted as though a trivial Seifert fibred space over the annulus with one twisted reflector boundary (and one twisted puncture corresponding to the block boundary) had been replaced by this block. In particular, this routine should not add the reflector boundary itself.

Parameters:
sfs the Seifert fibred space to adjust.
reflect true if this block is to be reflected, or false if it should be inserted directly.

Implements regina::NSatBlock.

void regina::NSatLayering::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.

void regina::NSatLayering::writeAbbr ( std::ostream &  out,
bool  tex = false 
) const [inline, virtual]

Writes an abbreviated name or symbol for this block to the given output stream.

This name should reflect the particular block type, but need not provide thorough details.

The output should be no more than a handful of characters long, and no newline should be written. In TeX mode, no leading or trailing dollar signs should be written.

Python:
The parameter out does not exist; standard output will be used.
Parameters:
out the output stream to which to write.
tex true if the output should be formatted for TeX, or false if it should be in plain text format.

Implements regina::NSatBlock.

static NSatLayering* regina::NSatLayering::isBlockLayering ( const NSatAnnulus annulus,
TetList avoidTets 
) [static]

Determines whether the given annulus is a boundary annulus for a block of this type (single layering).

This routine is a specific case of NSatBlock::isBlock(); see that routine for further details.

Parameters:
annulus the proposed boundary annulus that should form part of the new saturated block.
avoidTets the list of tetrahedra that should not be considered, and to which any new tetrahedra will be added.
Returns:
details of the saturated block if one was found, or null if none was found.


The documentation for this class was generated from the following file:
Copyright © 1999-2006, 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).