This class offers routines for constructing ready-made examples of knots and links.
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| static Link | unknot () |
| | Returns a zero-crossing diagram of the unknot.
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| static Link | monster () |
| | Returns the monster unknot, a 10-crossing diagram of the unknot that is difficult to untangle.
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| static Link | gordian () |
| | Returns Haken's Gordian unknot, a 141-crossing diagram of the unknot that is difficult to untangle.
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| static Link | trefoilLeft () |
| | Returns a three-crossing diagram of the left-hand trefoil.
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| static Link | trefoilRight () |
| | Returns a three-crossing diagram of the right-hand trefoil.
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| static Link | trefoil () |
| | Returns a three-crossing diagram of the right-hand trefoil.
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| static Link | figureEight () |
| | Returns a four-crossing diagram of the figure eight knot.
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| static Link | hopf () |
| | Returns a two-crossing diagram of the Hopf link.
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| static Link | whitehead () |
| | Returns a five-crossing diagram of the Whitehead link.
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| static Link | borromean () |
| | Returns a six-crossing diagram of the Borromean rings.
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| static Link | conway () |
| | Returns the 11-crossing Conway knot.
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| static Link | kinoshitaTerasaka () |
| | Returns the 11-crossing Kinoshita-Terasaka knot.
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| static Link | torus (int p, int q) |
| | Returns the (p,q) torus link.
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| static Link | gst () |
| | Returns a 48-crossing potential counterexample to the slice-ribbon conjecture, as described by Gompf, Scharlemann and Thompson.
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| static Link | chen () |
| | Returns a 20-crossing, 5-component counterexample to the 3-move conjecture, as proposed by Chen and proven to be a counterexample by Dabkowski and Przytycki.
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| static Link | virtualTrefoil () |
| | Returns a two-crossing diagram of the virtual trefoil.
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| static Link | kishino () |
| | Returns a four-crossing diagram of the Kishino knot.
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| static Link | gpv () |
| | Returns a four-crossing diagram of the Goussarov-Polyak-Viro virtual knot.
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| static SpatialLink | spatialTrefoil () |
| | Returns a simple and symmetric embedding in 3-space of the right-hand trefoil.
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| static SpatialLink | spatialHopf () |
| | Returns a simple embedding in 3-space of the Hopf link.
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| static SpatialLink | spatialBorromean () |
| | Returns a simple and symmetric embedding in 3-space of the Borromean rings.
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| static SpatialLink | cubicalUnknot () |
| | Returns a 3-dimensional embedding of the unknot that follows the edges of a cube.
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This class offers routines for constructing ready-made examples of knots and links.
These examples may be useful for testing new code, or for simply getting a feel for how Regina works.
The sample links offered here may prove especially useful in Regina's scripting interface, where working with pre-existing files is more complicated than in the GUI.
| static Link regina::ExampleLink::chen |
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Returns a 20-crossing, 5-component counterexample to the 3-move conjecture, as proposed by Chen and proven to be a counterexample by Dabkowski and Przytycki.
This link was proposed as a potential counterexample to the 3-move conjecture in "The 3-move conjecture for 5-braids", Qi Chen, Knots in Hellas '98, Proceedings of the International Conference on Knot Theory and its Ramifications, Series on Knots and Everything, Vol. 24, World Scientific, 2000, pp. 36-47.
It was proven to be a counterexample in "Burnside obstructions to
the Montesinos-Nakanishi 3-move conjecture", M. K. Dabkowski and J. H. Przytycki, Geometry and Topology 6 (2002), 335-360.
- Returns
- Chen's proposed (and since proven) 20-crossing counterexample to the 3-move conjecture.
| static Link regina::ExampleLink::gpv |
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Returns a four-crossing diagram of the Goussarov-Polyak-Viro virtual knot.
This is a knot whose group changes when we switch the upper and lower strands at each crossing (a behaviour that is impossible for classical knots and links).
Specifically: if we denote this knot K, then K.group() is isomorphic to the trefoil group; however, if we call K.changeAll() or K.rotate() then K.group() becomes isomorphic to the unknot group (i.e., the infinite cyclic group).
This is the rotation of virtual knot 4.73 in the Jeremy Green tables (where by "rotation" we mean flipping the diagram upside-down so that each crossing keeps its sign but switches its upper vs lower strands - in Green's terminology, this is the composition of both a vertical and a horizontal mirror image).
- Returns
- the Goussarov-Polyak-Viro virtual knot.
| static Link regina::ExampleLink::gst |
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Returns a 48-crossing potential counterexample to the slice-ribbon conjecture, as described by Gompf, Scharlemann and Thompson.
Specifically, this knot is Figure 2 from their paper "Fibered knots and potential counterexamples to the property 2R and
slice-ribbon conjectures", Geometry & Topology 14 (2010), 2305-2347.
- Returns
- the Gompf-Scharlemann-Thompson knot.
| static Link regina::ExampleLink::kishino |
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Returns a four-crossing diagram of the Kishino knot.
This is a non-trivial virtual knot that is the composition of two virtual unknots. It is a non-trivial virtual knot; however, it has the same group as the unknot, and it has trivial Jones polynomial.
This is virtual knot 4.55 in the Jeremy Green tables.
- Returns
- the Kishino knot.