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/**************************************************************************
* *
* Regina - A Normal Surface Theory Calculator *
* Computational Engine *
* *
* Copyright (c) 1999-2025, Ben Burton *
* For further details contact Ben Burton (bab@debian.org). *
* *
* This program is free software; you can redistribute it and/or *
* modify it under the terms of the GNU General Public License as *
* published by the Free Software Foundation; either version 2 of the *
* License, or (at your option) any later version. *
* *
* As an exception, when this program is distributed through (i) the *
* App Store by Apple Inc.; (ii) the Mac App Store by Apple Inc.; or *
* (iii) Google Play by Google Inc., then that store may impose any *
* digital rights management, device limits and/or redistribution *
* restrictions that are required by its terms of service. *
* *
* This program is distributed in the hope that it will be useful, but *
* WITHOUT ANY WARRANTY; without even the implied warranty of *
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU *
* General Public License for more details. *
* *
* You should have received a copy of the GNU General Public License *
* along with this program. If not, see <https://www.gnu.org/licenses/>. *
* *
**************************************************************************/
/*! \file link/examplelink.h
* \brief Offers several ready-made examples of knots and links.
*/
#ifndef __REGINA_EXAMPLELINK_H
#ifndef __DOXYGEN
#define __REGINA_EXAMPLELINK_H
#endif
#include "regina-core.h"
namespace regina {
class Link;
class SpatialLink;
/**
* 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.
*
* \ingroup link
*/
class ExampleLink {
public:
/**
* Returns a zero-crossing diagram of the unknot.
*
* \return the unknot.
*/
static Link unknot();
/**
* Returns the monster unknot, a 10-crossing diagram of the
* unknot that is difficult to untangle.
*
* \return the monster unknot.
*/
static Link monster();
/**
* Returns Haken's Gordian unknot, a 141-crossing diagram of the
* unknot that is difficult to untangle.
*
* \return the Gordian unknot.
*/
static Link gordian();
/**
* Returns a three-crossing diagram of the left-hand trefoil.
*
* \return the left-hand trefoil.
*/
static Link trefoilLeft();
/**
* Returns a three-crossing diagram of the right-hand trefoil.
* This returns the same knot as trefoil().
*
* \return the right-hand trefoil.
*/
static Link trefoilRight();
/**
* Returns a three-crossing diagram of the right-hand trefoil.
* This returns the same knot as trefoilRight().
*
* \return the right-hand trefoil.
*/
static Link trefoil();
/**
* Returns a four-crossing diagram of the figure eight knot.
*
* \return the figure eight knot.
*/
static Link figureEight();
/**
* Returns a two-crossing diagram of the Hopf link.
* This is the variant in which both crossings are positive.
*
* \return the Hopf link.
*/
static Link hopf();
/**
* Returns a five-crossing diagram of the Whitehead link.
*
* \return the Whitehead link.
*/
static Link whitehead();
/**
* Returns a six-crossing diagram of the Borromean rings.
*
* \return the Borromean rings.
*/
static Link borromean();
/**
* Returns the 11-crossing Conway knot.
*
* This is the reflection of \a K11n34 in the Knot Atlas, and is
* a mutant of the Kinoshita-Terasaka knot.
*
* \return the Conway knot.
*/
static Link conway();
/**
* Returns the 11-crossing Kinoshita-Terasaka knot.
*
* This is the reflection of \a K11n42 in the Knot Atlas, and is
* a mutant of the Conway knot. It has trivial Alexander polynomial.
*
* \return the kinoshita-Terasaka knot.
*/
static Link kinoshitaTerasaka();
/**
* Returns the (\a p,\a q) torus link.
*
* The parameters \a p and \a q must be non-negative, but they do
* not need to be coprime.
*
* All of the crossings in the resulting link will be positive.
*
* \param p the first parameter of the torus link; this must be
* strictly non-negative.
* \param q the second parameter of the torus link; this must
* also be strictly non-negative.
* \return the (\a p, \a q) torus link.
*/
static Link torus(int p, int q);
/**
* 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.
*
* \return the Gompf-Scharlemann-Thompson knot.
*/
static Link gst();
/**
* 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.
*
* \return Chen's proposed (and since proven) 20-crossing
* counterexample to the 3-move conjecture.
*/
static Link chen();
/**
* Returns a two-crossing diagram of the virtual trefoil.
* Both crossings will be positive.
*
* This is the mirror image of virtual knot 2.1 in the Jeremy Green
* tables (where by "mirror image" we mean switching the upper and
* lower strands in each crossing - Green calls this a _vertical_
* mirror image).
*
* \return the virtual trefoil.
*/
static Link virtualTrefoil();
/**
* 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.
*
* \return the Kishino knot.
*/
static Link kishino();
/**
* 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 \a 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).
*
* \return the Goussarov-Polyak-Viro virtual knot.
*/
static Link gpv();
/**
* Returns a simple and symmetric embedding in 3-space of the
* right-hand trefoil.
*
* \return the right-hand trefoil.
*/
static SpatialLink spatialTrefoil();
/**
* Returns a simple embedding in 3-space of the Hopf link.
*
* \return the Hopf link.
*/
static SpatialLink spatialHopf();
/**
* Returns a simple and symmetric embedding in 3-space of the
* Borromean rings.
*
* \return the Borromean rings.
*/
static SpatialLink spatialBorromean();
/**
* Returns a 3-dimensional embedding of the unknot that follows the
* edges of a cube.
*
* This is not a planar embedding: instead it follows a cycle through
* 6 of the 12 edges of the cube, making use of all three dimensions.
*
* \return an unknot embedded in the edges of a cube.
*/
static SpatialLink cubicalUnknot();
// Make this class non-constructible.
ExampleLink() = delete;
};
} // namespace regina
#endif
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