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What is the shortest possible length of a cycle in the 14-regular* graph of the BCC lattice** with a given knot type?
Here are my current suggestions for some of the simplest ones. All models created using zometool construction set.

For other cubic lattices, vist Andrew Rechnitzer's page on this topic.


Knot type Trefoil Figure-eight Cinquefoil Three-twist 61 62 63 819
Length1316181822222224
Proven
to be
optimal?
CommentFive
optimal
isometry
classes
No shorter
cycle with
geometric
diameter
less than
6 (confer
coordinates)
Model
Successive
coordinates
0 2 0
2 2 0
2 2 2
2 4 2
4 4 2
4 2 2
3 1 1
1 1 1
1 3 1
3 3 1
3 3 3
1 3 3
0 2 2
1 1 1
3 1 1
5 1 1
5 1 3
3 1 3
2 2 2
2 2 0
2 4 0
2 4 2
4 4 2
4 2 2
4 0 2
4 0 0
4 2 0
3 3 1
1 3 1
0 2 0
2 2 0
3 1 1
5 1 1
5 3 1
3 3 1
3 3 3
1 3 3
1 3 1
1 1 1
2 0 0
4 0 0
4 2 0
4 2 2
4 4 2
2 4 2
2 2 2
0 2 2
0 2 0
2 2 0
2 2 2
4 2 2
4 4 2
2 4 2
0 4 2
0 4 4
2 4 4
3 3 3
3 3 1
3 1 1
1 1 1
1 3 1
1 5 1
1 5 3
1 3 3
0 2 2
4 0 0
6 0 0
6 2 0
6 2 2
6 4 2
4 4 2
4 2 2
4 0 2
2 0 2
2 2 2
3 3 3
5 3 3
5 3 1
7 3 1
7 1 1
5 1 1
4 2 0
2 2 0
1 1 1
1 1 3
3 1 3
3 1 1
2 0 2
2 0 4
2 2 4
1 3 5
1 5 5
1 5 3
1 3 3
1 3 1
3 3 1
3 3 3
3 3 5
3 1 5
1 1 5
1 1 3
3 1 3
4 2 4
4 4 4
2 4 4
0 4 4
0 4 2
2 4 2
2 2 2
1 1 1
3 1 1
4 2 2
4 4 2
4 4 0
4 2 0
2 2 0
2 2 2
2 2 4
2 4 4
2 4 2
2 4 0
0 4 0
0 4 2
1 3 3
3 3 3
5 3 3
5 3 1
3 3 1
3 5 1
1 5 1
1 3 1
2 0 2
4 0 2
5 1 3
5 3 3
3 3 3
1 3 3
1 1 3
3 1 3
3 1 1
5 1 1
5 3 1
3 3 1
3 5 1
3 5 3
4 4 4
4 2 4
4 2 2
4 2 0
4 4 0
4 4 2
2 4 2
2 4 4
2 2 4
2 2 2

*Apparently, it is conventional to consider the 8-regular graph with only the shortest edges for this lattice.
In my humble opinion, it is also natural to consider the dual of the tesselation of space using truncated octahedra, which is what I have done here.

**Please note that the packing density given on the linked web page is incorrect for BCC (as of 14.07.16).