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).

• The knot atlas developed (mostly) by Scott Morrison and Dror Bar-Natan.
• Knotplot developed by Rob Scharein.

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 Stu 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

More coming soon - maybe some composite knots.