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 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.