$12^{1}_{64}$ - Minimal pinning sets
Pinning sets for 12^1_64 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_64 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 280
of which optimal: 3
of which minimal: 6
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.04505
on average over minimal pinning sets: 2.51111
on average over optimal pinning sets: 2.46667
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 7, 8, 11}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {1, 3, 4, 7, 8}
5
[2, 2, 2, 3, 3]
2.40
C (optimal)
• {1, 2, 3, 7, 8}
5
[2, 2, 2, 3, 4]
2.60
a (minimal)
• {1, 3, 6, 7, 9, 11}
6
[2, 2, 2, 3, 3, 3]
2.50
b (minimal)
• {1, 3, 4, 6, 7, 9}
6
[2, 2, 2, 3, 3, 3]
2.50
c (minimal)
• {1, 2, 3, 6, 7, 9}
6
[2, 2, 2, 3, 3, 4]
2.67
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
3
0
0
2.47
6
0
3
18
2.71
7
0
0
58
2.9
8
0
0
84
3.05
9
0
0
70
3.16
10
0
0
34
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
3
3
274
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,3,4,2],[0,1,4,0],[0,5,6,1],[1,7,5,2],[3,4,8,6],[3,5,9,9],[4,9,8,8],[5,7,7,9],[6,8,7,6]]
PD code (use to draw this loop with SnapPy): [[11,20,12,1],[10,13,11,14],[19,12,20,13],[1,15,2,14],[18,9,19,10],[15,9,16,8],[2,8,3,7],[4,17,5,18],[16,5,17,6],[3,6,4,7]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (14,3,-15,-4)(1,4,-2,-5)(5,20,-6,-1)(6,13,-7,-14)(16,7,-17,-8)(17,10,-18,-11)(8,11,-9,-12)(12,19,-13,-20)(2,15,-3,-16)(9,18,-10,-19)
Edge permutation $\epsilon=$ (-1,1)(-2,2)(-3,3)(-4,4)(-5,5)(-6,6)(-7,7)(-8,8)(-9,9)(-10,10)(-11,11)(-12,12)(-13,13)(-14,14)(-15,15)(-16,16)(-17,17)(-18,18)(-19,19)(-20,20)
Face permutation $\varphi=(\sigma\epsilon)^{-1}=$ (-1,-5)(-2,-16,-8,-12,-20,5)(-3,14,-7,16)(-4,1,-6,-14)(-9,-19,12)(-10,17,7,13,19)(-11,8,-17)(-13,6,20)(-15,2,4)(-18,9,11)(3,15)(10,18)
Loop annotated with half-edges
12^1_64 annotated with half-edges