# 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. After accumulating all this data I thought I would put it all in one place where it would be relatively easily accessible. Clicking on a knot below will take you to some minimal representations of that knot on these three lattices.

Buks hand-coded embeddings of the knots (up to 7 crossings) in the simple cubic lattice and then minimised them using the BFACF algorithm (see this for example). I coded up to 5 crossings by hand on the FCC and BCC and then minimised them using BFACF moves. After that I realised that I it was far easier to simply project Buks' simple cubic lattice embeddings into a sublattice of the FCC and BCC and then minimised them.

To find all the other embeddings (8 crossings and up) I used the minimal embeddings in the simple cubic lattice that appear in the paper by Scharein et. al. and then mapped them to sublattices of the FCC and BCC (simple perl scripts helped to automate the whole process). Finally the embeddings were minimised using the GAS algorithm (or a simple variant of it).

For way way more information on knots I strongly recommend taking a look at

Note that the knot images below are taken from here.

I should really learn some php instead of having all these files. I'll get around to it.

Update February 27th 2011 - I have added the symmetry classes for all the knots with 7 crossings or fewer (corrected a couple of earlier errors). They are encoded as an oriented list of edges (and so have up to 48 symmetries). The edges are labelled as per here (in hexadecimal for the FCC). I have tentatively added the files for the 8 crossing knots - a few knots are missing a few classes, but I should have them patched in a week or so.

Update March 1st 2011 - I have fixed up the symmetry classes for all of the 8 crossing knots. Enjoy.

## Knots with 7 crossings or fewer

 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 For Stu9_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