# Knots and their symmetries

I have been working through the standard list of small prime knots, with the aim of creating elegant and, where possible, symmetric three-dimensional realizations. Here you will find my versions of the knots described in terms of 2D diagrams, lists of coordinates of points along the curve, Fourier series, .scad files and .stl files. I follow the nomenclature used in Knotinfo and Knotserver.

 Knot 31 Trefoil Knot 41 Figure-eight Knot 51 Cinquefoil Knot 52 Three-twist Knot 61 Stevedore Knot 62 Knot 63 Knot 71 Knot 72 Knot 73 Knot 74 Eternal Knot 75 Knot 76 Knot 77 Knot 81 Knot 82 Knot 83 Knot 84 Knot 85 Knot 86 Knot 87 Knot 88 Knot 89 Knot 810 Knot 811 Knot 812 Knot 813 Knot 814 Knot 815 Knot 816 Knot 817 Knot 818 Knot 819 Knot 820 Knot 821 Knot 12a1202 Knot 15331

Many small knots have symmetric three-dimensional realizations. I have therefore sought to offer these when possible. On the pages devoted to individual knots, I name symmetries (D2, D4 etc) when I have found them. Note that a knot may have various symmetries witnessed by different realizations. By "symmetry" here what I really mean is a conjugacy class of finite subgroups of the isometry group of S3, but I think most of us would prefer to have a list of descriptions relating to the rather arbitrary names I have used:

• D2(p) means invariance under a rotation of 180° around an axis, preserving orientation of the knot,
• D2(r) means invariance under a rotation of 180° around an axis, reversing orientation of the knot,
• D4, D6, D8, D10, D14 etc mean invariance under a rotation of 180°, 120°, 90°, 72°, (360/7)° around an axis (preserving orientation), together with invariance under a rotation of 180° around an axis at right angles to the first (reversing orientation),
• Z2, Z4, Z6 etc mean invariance under an orientation-preserving operation consisting of a rotation of 180°, 90°, 60° around an axis followed (or preceded) by reflection in a plane perpendicular to that axis,
• I2 means invariance under inversion in a sphere followed by a rotation of 180° about an axis through the centre of the sphere (reversing orientation),
• I8 means invariance under inversion in a sphere followed by a rotation of 180°, as in I2, together with invariance under a rotation of 90° (about an axis through the centre of the sphere orthogonal to the first) followed by reflection in the plane perpendicular to the second axis through the center of the sphere, as in Z4.

All knots with D2(r) symmetry are strongly invertible. Of the knots here, 41, 63, 83, 89, 812, 818 and 12a1202 are fully amphichiral; 817 and 15331 are negatively amphichiral. I believe that 15331 is prime.

28.3.20