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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/>. *
* *
**************************************************************************/
#include "link/examplelink.h"
#include "link/link.h"
#include "link/spatiallink.h"
// An right-handed orthogonal basis in R^3 that breaks symmetry.
// All three basis elements have the same length.
// We use this to build spatial links whose structure is easily visible when
// seen from any of the "pure" axis directions.
static constexpr regina::SpatialLink::Node orth[3] = {
regina::SpatialLink::Node{ 6.0, 3.0, -2.0 },
regina::SpatialLink::Node{ -2.0, 6.0, 3.0 },
regina::SpatialLink::Node{ 3.0, -2.0, 6.0 }
};
namespace regina {
Link ExampleLink::unknot() {
return { 1 };
}
Link ExampleLink::trefoilLeft() {
return Link::fromData({ -1, -1, -1 }, { 1, -2, 3, -1, 2, -3 });
}
Link ExampleLink::trefoilRight() {
return Link::fromData({ +1, +1, +1 }, { 1, -2, 3, -1, 2, -3 });
}
Link ExampleLink::trefoil() {
return Link::fromData({ +1, +1, +1 }, { 1, -2, 3, -1, 2, -3 });
}
Link ExampleLink::figureEight() {
return Link::fromData({ +1, +1, -1, -1 }, { -1, 2, -3, 4, -2, 1, -4, 3});
}
Link ExampleLink::hopf() {
return Link::fromData({ +1, +1 }, { 1, -2 }, { -1, 2 });
}
Link ExampleLink::whitehead() {
return Link::fromData({ -1, -1, +1, +1, -1 },
{ 1, -2, 5, -4, 3, -5 }, { -1, 2, -3, 4 });
}
Link ExampleLink::borromean() {
return Link::fromData({ +1, -1, +1, -1, +1, -1 },
{ 1, -2, 4, -5 }, { 3, -4, 6, -1 }, { 5, -6, 2, -3 });
}
Link ExampleLink::monster() {
return Link::fromData({ -1, -1, +1, -1, +1, +1, +1, +1, +1, -1 },
{ 1, -2, 3, 4, 5, -6, 7, -8, 9, -3, 10, -1, 2, -10, -4, -9, 8, -5,
6, -7 });
}
Link ExampleLink::kinoshitaTerasaka() {
return Link::fromData({ -1, -1, +1, -1, -1, +1, -1, +1, +1, +1, -1 },
{ 1, -2, 3, 4, -5, -6, 2, -7, -8, 9, -4, 5, -10, 8, -9, 10, 11,
-1, 6, -3, 7, -11 });
}
Link ExampleLink::conway() {
return Link::fromData({ +1, -1, -1, -1, -1, -1, -1, +1, +1, +1, +1 },
{ 1, -2, 3, 4, -5, -6, 7, -1, 8, -3, 6, -7, 2, -8, -9, 10, -11,
9, -4, 5, -10, 11 });
}
Link ExampleLink::gordian() {
return Link::fromData(
{ +1, -1, +1, +1, -1, +1, +1, -1, -1, +1, +1, -1, +1, +1, +1,
-1, -1, -1, -1, -1, -1, +1, -1, -1, -1, -1, -1, -1, -1, -1,
-1, -1, -1, +1, -1, +1, -1, +1, +1, -1, +1, +1, -1, -1, -1,
-1, +1, +1, -1, -1, +1, -1, +1, -1, -1, -1, -1, -1, +1, -1,
-1, +1, -1, -1, +1, -1, -1, -1, -1, -1, -1, +1, +1, +1, -1,
+1, +1, +1, +1, -1, -1, +1, -1, +1, +1, +1, -1, -1, +1, -1,
-1, +1, -1, +1, +1, +1, +1, +1, -1, +1, +1, +1, +1, -1, -1,
-1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, +1, -1, -1,
+1, -1, -1, -1, -1, -1, +1, -1, -1, -1, -1, -1, -1, +1, -1,
+1, +1, +1, -1, +1, -1 },
{ -1, -2, 3, -4, 5, 6, 7, 8, -9, -10, 11, 12, 13, -14, 15, -16, 17,
18, -19, 20, -21, -7, 22, 23, 24, -25, -26, -27, 28, -29, 30, 31,
-32, -33, -34, 35, 36, -37, -38, -39, -40, -41, 42, 43, -44, 45,
46, 47, -48, -49, -50, 38, -51, 52, -53, -54, -55, -42, -56, 57, 58,
-46, -59, -60, -61, -62, 63, -64, -65, -66, -67, 68, -35, -69, 70,
-71, -72, 73, 29, -74, -31, -75, -76, 77, 69, -36, -68, -78, -79,
-80, 64, -63, -81, -82, 60, -83, -47, 84, -57, -85, -43, 55, -86,
-87, -52, -88, 37, -89, 49, -90, -84, -58, -45, 44, 85, 56, -91,
-92, 39, 51, 88, -70, -77, -93, 33, -94, 74, -30, -73, 95, 27, -96,
-97, 98, -23, 99, -8, 21, 100, -101, -18, 102, -103, -15, -104, 105,
-12, 106, -107, 9, -99, -22, 108, -5, -109, 110, 2, -111, -112, 113,
-114, -28, -95, 72, 115, -116, 75, 32, 94, 117, 78, -118, 66, -119,
120, 121, 83, 59, 10, 107, -122, -123, 124, 103, 16, 125, -110, -3,
126, 127, -98, -24, 54, 86, -128, 40, 92, 129, 90, 48, -121, -130,
62, 81, -131, 104, 14, 132, -133, -124, -134, 19, 101, 135, -6,
-108, -127, 136, 137, 114, -113, -138, 139, -126, 4, 109, -135,
-100, -20, 140, 123, 133, -132, -13, -105, 131, 141, 79, 118, 67,
89, 50, -129, 91, 41, 128, 87, 53, 25, 97, -136, -139, 111, 1, -125,
-17, -102, 134, -140, 122, -106, -11, 82, 61, 130, -120, 119, 65,
80, -141, -117, 34, 93, 76, 116, -115, 71, 26, 96, -137, 138, 112 });
}
Link ExampleLink::torus(int p, int q) {
Link ans;
ans.insertTorusLink(p, q);
return ans;
}
Link ExampleLink::gst() {
return Link::fromData(
{ -1, -1, -1, +1, +1, -1, -1, -1, -1, -1, +1, -1, +1, +1, +1, -1,
-1, -1, +1, -1, +1, +1, +1, -1, +1, -1, +1, +1, +1, -1, -1, +1,
-1, -1, +1, +1, +1, +1, +1, +1, -1, -1, -1, -1, +1, +1, +1, +1 },
{ -1, -2, -3, -4, 5, 6, 7, 8, -9, -10, 11, -12,
-13, -14, 15, 16, 2, -7, 17, 9, -18, 19, -20, -21,
-22, 23, 24, 1, -8, -17, 10, 18, 25, -26, -27, -28,
29, 30, -31, -32, -33, 27, 21, 13, 34, -5, 4, -15,
-23, -29, 32, 35, 36, -37, -38, 33, 28, 22, 14, -34,
-39, -11, -19, -25, 40, -41, -42, 43, 44, 31, -30, -24,
-16, 3, -6, 39, 12, 20, 26, -40, 45, 46, -43, -36,
37, 42, -46, -47, 48, -44, -35, 38, 41, -45, 47, -48 });
}
Link ExampleLink::chen() {
return Link::fromData(
{ -1, -1, -1, -1,
+1, +1, +1, +1,
+1, +1, +1, +1,
-1, -1, -1, -1,
+1, +1, +1, +1 },
{ 1, -2, 6, 10, -14, 15, -19, -8 },
{ 2, -3, 7, 11, -15, 16, -20, -5 },
{ 3, -4, 8, 12, -16, 13, -17, -6 },
{ 4, -1, 5, 9, -13, 14, -18, -7 },
{ 17, -10, 18, -11, 19, -12, 20, -9 });
}
Link ExampleLink::virtualTrefoil() {
return Link::fromData({ +1, +1 }, { 1, -2, -1, 2 });
}
Link ExampleLink::kishino() {
return Link::fromData({ -1, +1, +1, -1 }, { 1, -2, -1, 2, -3, 4, 3, -4 });
}
Link ExampleLink::gpv() {
return Link::fromData({ -1, -1, -1, -1 }, { 1, -2, -1, 2, 3, -4, -3, 4 });
}
SpatialLink ExampleLink::spatialTrefoil() {
// Note: we could happily remove the second and fourth columns; the
// reason we keep them is to add a bend to the arcs and make everything
// look less "sticky".
return SpatialLink({{
// CORNER INNER ARC CORNER OUTER ARC
{0.0, 2.0, 2.0}, {2.5, 1.5, 1.5}, {5.0, 2.0, 2.0}, {4.2, 4.2, 2.5},
{2.0, 5.0, 3.0}, {1.5, 2.5, 3.5}, {2.0, 0.0, 3.0}, {2.5, 0.8, 0.8},
{3.0, 3.0, 0.0}, {3.5, 3.5, 2.5}, {3.0, 3.0, 5.0}, {0.8, 2.5, 4.2}}});
}
SpatialLink ExampleLink::spatialHopf() {
// TODO: Is this the positive or negative linking number variant?
// Work this out, and document it.
static const double r = std::sqrt(double(6.0));
// We build the two components from equilateral triangles.
// The first component has centroid (2,2,2) and side length 6*sqrt(2).
// The second component has centroid (0,3,3).
//
// We add intermediate points to make each component into a regular hexagon
// (but these are not actually necessary; they're just for aesthetics).
//
// Finally: we shift the second component by (-1,0.5,0.5) so the
// two components are a little further apart.
SpatialLink ans({
{ {6.0, 0.0, 0.0}, // a
{4.0, 4.0, -2.0}, // (2a+2b-c)/3
{0.0, 6.0, 0.0}, // b
{-2.0, 4.0, 4.0}, // (2b+2c-a)/3
{0.0, 0.0, 6.0}, // c
{4.0, -2.0, 4.0} }, // (2a+2c-b)/3
{ {1.0+r, 2.5+r, 2.5+r},
{3.0, 1.5, 1.5},
{1.0-r, 2.5-r, 2.5-r},
{-3.0-r, 4.5-r, 4.5-r},
{-5.0, 5.5, 5.5},
{-3.0+r, 4.5+r, 4.5+r} }});
return ans;
}
SpatialLink ExampleLink::spatialBorromean() {
SpatialLink ans({
{ orth[0] * 2.0 + orth[1], orth[0] * -2.0 + orth[1],
orth[0] * -2.0 + orth[1] * -1.0, orth[0] * 2.0 + orth[1] * -1.0 },
{ orth[1] * 2.0 + orth[2], orth[1] * -2.0 + orth[2],
orth[1] * -2.0 + orth[2] * -1.0, orth[1] * 2.0 + orth[2] * -1.0 },
{ orth[2] * 2.0 + orth[0], orth[2] * -2.0 + orth[0],
orth[2] * -2.0 + orth[0] * -1.0, orth[2] * 2.0 + orth[0] * -1.0 }});
ans.refine(4);
return ans;
}
SpatialLink ExampleLink::cubicalUnknot() {
return SpatialLink({{
{0.0, 0.0, 0.0}, orth[1], orth[0] + orth[1],
orth[0] + orth[1] + orth[2], orth[0] + orth[2], orth[2] }});
}
} // namespace regina
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