Regina Calculation Engine
Classes | Enumerations | Functions
Knots and Links

Knots and links in the 3-sphere. More...

Classes

class  regina::ExampleLink
 This class offers routines for constructing ready-made examples of knots and links. More...
 
class  regina::Laurent< T >
 
Represents a single-variable Laurent polynomial with coefficients of type T. More...
 
class  regina::Laurent2< T >
 
Represents a Laurent polynomial in the two variables x, y with coefficients of type T. More...
 
class  regina::Triangulation< dim >
 
A dim-dimensional triangulation, built by gluing together dim-dimensional simplices along their (dim-1)-dimensional facets. More...
 
class  regina::StrandRef
 A reference to one of the two strands of a link that pass each other at a crossing. More...
 
class  regina::Crossing
 Represents a single crossing in a link diagram. More...
 
class  regina::Link
 Represents a directed knot or link in the 3-sphere. More...
 
class  regina::CrossingIterator
 
Iterates through all crossings of a link. More...
 
class  regina::ArcIterator
 
Iterates through all directed arcs of a knot or link. More...
 
class  regina::ModelLinkGraphArc
 A reference to an outgoing edge from a node of a model graph for a knot or link. More...
 
class  regina::ModelLinkGraphNode
 Represents a single node in a model graph for a knot or link. More...
 
class  regina::ModelLinkGraph
 Represents an undirected 4-valent planar graph with a specific planar embedding. More...
 
class  regina::ModelLinkGraphCells
 Describes the cellular decomposition of the sphere that is induced by a given planar 4-valent graph. More...
 
class  regina::Tangle
 Represents a 2-tangle in the 3-ball. More...
 
class  regina::XMLLinkReader
 An XML packet reader that reads a single knot or link. More...
 
class  regina::XMLLinkCrossingsReader
 Helper class that reads the XML element containing basic information about the crossings of a knot or link. More...
 
class  regina::XMLLinkConnectionsReader
 Helper class that reads the XML element containing information on connections between crossings of a knot or link. More...
 
class  regina::XMLLinkComponentsReader
 Helper class that reads the XML element containing information about the individual components of a link. More...
 

Enumerations

enum  regina::Framing { regina::FRAMING_SEIFERT = 1, regina::FRAMING_BLACKBOARD = 2 }
 Indicates one of the standard framings of a knot or link. More...
 

Functions

std::ostream & regina::operator<< (std::ostream &out, const StrandRef &s)
 Writes a depiction of the given strand reference to the given output stream. More...
 
std::ostream & regina::operator<< (std::ostream &out, const ModelLinkGraphArc &a)
 Writes a depiction of the given arc reference to the given output stream. More...
 

Detailed Description

Knots and links in the 3-sphere.

Enumeration Type Documentation

◆ Framing

Indicates one of the standard framings of a knot or link.

Here a framing refers to a choice of normal vector field along the knot or link. Equivalently, a framing refers to a choice of longitude on the torus bounding each component of the link.

Enumerator
FRAMING_SEIFERT 

Indicates the Seifert framing, which is defined algebraically and is independent of the knot/link projection.

For each component of the link, draw a Seifert surface (i.e., an orientable surface embedded in the 3-sphere that is bounded by the corresponding knot). The Seifert framing is the vector field that points into the corresponding surface.

Equivalently, for each component of the link, the Seifert framing chooses the unique longitude for the corresponding knot that is trivial in the homology of the knot complement.

FRAMING_BLACKBOARD 

Indicates the blackboard framing, which is specific to the knot/link projection.

For the blackboard framing, the normal vector field stays within the projection plane. Equivalently, the blackboard framing chooses longitudes whose projections do not intersect the original link diagram.

Function Documentation

◆ operator<<() [1/2]

std::ostream & regina::operator<< ( std::ostream &  out,
const ModelLinkGraphArc a 
)
inline

Writes a depiction of the given arc reference to the given output stream.

Parameters
outthe output stream to which to write.
athe arc reference to write.
Returns
a reference to the given output stream.

◆ operator<<() [2/2]

std::ostream & regina::operator<< ( std::ostream &  out,
const StrandRef s 
)
inline

Writes a depiction of the given strand reference to the given output stream.

The reference will be written in the form ^n or _n, denoting the upper or lower strand at crossing n respectively. For example, the upper strand of crossing 7 will be written as ^7.

Parameters
outthe output stream to which to write.
sthe reference to write.
Returns
a reference to the given output stream.

Copyright © 1999-2021, The Regina development team
This software is released under the GNU General Public License, with some additional permissions; see the source code for details.
For further information, or to submit a bug or other problem, please contact Ben Burton (bab@maths.uq.edu.au).