$12^{1}_{322}$ - Minimal pinning sets
Pinning sets for 12^1_322 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_322 Pinning data
Pinning number of this loop: 4
Total number of pinning sets: 288
of which optimal: 1
of which minimal: 2
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.03457
on average over minimal pinning sets: 2.375
on average over optimal pinning sets: 2.25
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 6, 9}
4
[2, 2, 2, 3]
2.25
a (minimal)
• {1, 2, 4, 7, 8, 9}
6
[2, 2, 2, 3, 3, 3]
2.50
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
1
0
0
2.25
5
0
0
8
2.58
6
0
1
28
2.78
7
0
0
61
2.93
8
0
0
80
3.05
9
0
0
66
3.15
10
0
0
33
3.23
11
0
0
9
3.29
12
0
0
1
3.33
Total
1
1
286
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,2],[0,1,5,3],[0,2,6,7],[0,7,7,1],[1,8,6,2],[3,5,9,9],[3,8,4,4],[5,7,9,9],[6,8,8,6]]
PD code (use to draw this loop with SnapPy): [[13,20,14,1],[12,5,13,6],[19,4,20,5],[14,4,15,3],[1,7,2,6],[18,11,19,12],[15,11,16,10],[2,7,3,8],[8,17,9,18],[16,9,17,10]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (20,11,-1,-12)(16,1,-17,-2)(9,2,-10,-3)(3,18,-4,-19)(19,4,-20,-5)(13,6,-14,-7)(7,12,-8,-13)(15,8,-16,-9)(5,14,-6,-15)(10,17,-11,-18)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,16,8,12)(-2,9,-16)(-3,-19,-5,-15,-9)(-4,19)(-6,13,-8,15)(-7,-13)(-10,-18,3)(-11,20,4,18)(-12,7,-14,5,-20)(-17,10,2)(1,11,17)(6,14)
Loop annotated with half-edges
12^1_322 annotated with half-edges