Minimal knots on cubic lattices
As part of work (see this, this and this) on computing knotting probabilities Buks van Rensburg and I computed minimal knotted polygons on three cubic lattices: simple cubic, face-centred cubic and body-centred cubic. This adds to work done on the cubic lattice by Rob Scharein, Kai Ishihara, Javier Arsuaga, Yuanan Diao, Koya Shimokawa and Mariel Vazquez.
Clicking on a knot below will take you to its minimal representations on the simple cubic (SC), face-centred cubic (FCC), and body-centred cubic (BCC) lattices, with an interactive 3D viewer you can rotate, pan, and zoom. Data files with coordinates and symmetry classes are provided for each lattice.
For more information on knots I recommend:
- The Knot Atlas by Scott Morrison and Dror Bar-Natan.
- Knotplot by Rob Scharein.
Knot images below are from the Rolfsen knot table mosaic.
The 7 or fewer crossing knots
3_1 |
4_1 |
5_1 |
5_2 |
6_1 |
6_2 |
6_3 |
7_1 |
7_2 |
7_3 |
7_4 |
7_5 |
7_6 |
7_7 |
The 8 crossing knots
8_1 |
8_2 |
8_3 |
8_4 |
8_5 |
8_6 |
8_7 |
8_8 |
8_9 |
8_10 |
8_11 |
8_12 |
8_13 |
8_14 |
8_15 |
8_16 |
8_17 |
8_18 |
8_19 |
8_20 |
8_21 |
The 9 crossing knots
9_1 |
9_2 |
9_3 |
9_4 |
9_5 |
9_6 |
9_7 |
9_8 |
9_9 |
9_10 |
9_11 |
9_12 |
9_13 |
9_14 |
9_15 |
9_16 |
9_17 |
9_18 |
9_19 |
9_20 |
9_21 |
9_22 |
9_23 |
9_24 |
9_25 |
9_26 |
9_27 |
9_28 |
9_29 |
9_30 |
9_31 |
9_32 |
9_33 |
9_34 |
9_35 |
9_36 |
9_37 |
9_38 |
9_39 |
9_40 |
9_41 |
9_42 |
9_43 |
9_44 |
9_45 |
9_46 |
9_47 |
9_48 |
9_49 |
The 10 crossing knots
10_1 |
10_2 |
10_3 |
10_4 |
10_5 |
10_6 |
10_7 |
10_8 |
10_9 |
10_10 |
10_11 |
10_12 |
10_13 |
10_14 |
10_15 |
10_16 |
10_17 |
10_18 |
10_19 |
10_20 |
10_21 |
10_22 |
10_23 |
10_24 |
10_25 |
10_26 |
10_27 |
10_28 |
10_29 |
10_30 |
10_31 |
10_32 |
10_33 |
10_34 |
10_35 |
10_36 |
10_37 |
10_38 |
10_39 |
10_40 |
10_41 |
10_42 |
10_43 |
10_44 |
10_45 |
10_46 |
10_47 |
10_48 |
10_49 |
10_50 |
10_51 |
10_52 |
10_53 |
10_54 |
10_55 |
10_56 |
10_57 |
10_58 |
10_59 |
10_60 |
10_61 |
10_62 |
10_63 |
10_64 |
10_65 |
10_66 |
10_67 |
10_68 |
10_69 |
10_70 |
10_71 |
10_72 |
10_73 |
10_74 |
10_75 |
10_76 |
10_77 |
10_78 |
10_79 |
10_80 |
10_81 |
10_82 |
10_83 |
10_84 |
10_85 |
10_86 |
10_87 |
10_88 |
10_89 |
10_90 |
10_91 |
10_92 |
10_93 |
10_94 |
10_95 |
10_96 |
10_97 |
10_98 |
10_99 |
10_100 |
10_101 |
10_102 |
10_103 |
10_104 |
10_105 |
10_106 |
10_107 |
10_108 |
10_109 |
10_110 |
10_111 |
10_112 |
10_113 |
10_114 |
10_115 |
10_116 |
10_117 |
10_118 |
10_119 |
10_120 |
10_121 |
10_122 |
10_123 |
10_124 |
10_125 |
10_126 |
10_127 |
10_128 |
10_129 |
10_130 |
10_131 |
10_132 |
10_133 |
10_134 |
10_135 |
10_136 |
10_137 |
10_138 |
10_139 |
10_140 |
10_141 |
10_142 |
10_143 |
10_144 |
10_145 |
10_146 |
10_147 |
10_148 |
10_149 |
10_150 |
10_151 |
10_152 |
10_153 |
10_154 |
10_155 |
10_156 |
10_157 |
10_158 |
10_159 |
10_160 |
10_161 |
10_162 |
10_163 |
10_164 |
10_165 |