Regina Calculation Engine
A degenerate zero-tetrahedron saturated block that corresponds to attaching a Mobius band to a single annulus boundary. More...
|NSatMobius (const NSatMobius &cloneMe)|
|Constructs a clone of the given block structure. |
|int||position () const|
|Describes how the Mobius band is attached to the boundary annulus. |
|virtual NSatBlock *||clone () 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 NSatMobius *||isBlockMobius (const NSatAnnulus &annulus, TetList &avoidTets)|
|Determines whether the given annulus is a boundary annulus for a block of this type (Mobius band). |
A degenerate zero-tetrahedron saturated block that corresponds to attaching a Mobius band to a single annulus boundary.
This is a degenerate case of the layered solid torus (see the class NSatLST), where instead of joining a solid torus to an annulus boundary we join a Mobius band. The Mobius band can be thought of as a zero-tetrahedron solid torus with two boundary faces, which in fact are opposite sides of the same face. By attaching a zero-tetrahedron Mobius band to an annulus boundary, we are effectively joining the two faces of the annulus together.
The meridinal disc of this zero-tetrahedron solid torus meets the three edges of the annulus in 1, 1 and 2 places, so it is in fact a degenerate (1,1,2) layered solid torus. Note that the weight 2 edge is the boundary edge of the Mobius strip.
|regina::NSatMobius::NSatMobius||(||const NSatMobius &||cloneMe||)||
Constructs a clone of the given block structure.
|cloneMe||the block structure to clone.|
|virtual void regina::NSatMobius::adjustSFS||(||NSFSpace &||sfs,|
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space should be adjusted as though an ordinary solid torus (base orbifold a disc, no twists or exceptional fibres) had been replaced by this block.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.
|sfs||the Seifert fibred space to adjust.|
|NSatBlock * regina::NSatMobius::clone||(||)|| const
|static NSatMobius* regina::NSatMobius::isBlockMobius||(||const NSatAnnulus &||annulus,|
Determines whether the given annulus is a boundary annulus for a block of this type (Mobius band).
This routine is a specific case of NSatBlock::isBlock(); see that routine for further details.
|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.|
nullif none was found.
|int regina::NSatMobius::position||(||)|| const
Describes how the Mobius band is attached to the boundary annulus.
The class notes discuss the weight two edge of the Mobius band (or equivalently the boundary edge of the Mobius band). The return value of this routine indicates which edge of the boundary annulus this weight two edge is joined to.
In the NSatAnnulus class notes, the three edges of the annulus are denoted vertical, horizontal and boundary, and the vertices of each face are given markings 0, 1 and 2.
The return value of this routine takes the value 0, 1 or 2 as follows:
|virtual void regina::NSatMobius::writeAbbr||(||std::ostream &||out,|
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.
|out||the output stream to which to write.|
|virtual void regina::NSatMobius::writeTextShort||(||std::ostream &||out||)|| const
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.
|out||the output stream to which to write.|