$12^{1}_{94}$ - Minimal pinning sets
Pinning sets for 12^1_94 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_94 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 192
of which optimal: 2
of which minimal: 2
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.96906
on average over minimal pinning sets: 2.2
on average over optimal pinning sets: 2.2
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 10, 11}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 2, 3, 9, 11}
5
[2, 2, 2, 2, 3]
2.20
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
2
0
0
2.2
6
0
0
13
2.54
7
0
0
36
2.78
8
0
0
55
2.95
9
0
0
50
3.09
10
0
0
27
3.19
11
0
0
8
3.27
12
0
0
1
3.33
Total
2
0
190
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 3, 4, 5, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,4,5],[0,5,6,6],[0,7,8,9],[0,5,1,1],[1,4,9,2],[2,7,7,2],[3,6,6,8],[3,7,9,9],[3,8,8,5]]
PD code (use to draw this loop with SnapPy): [[3,20,4,1],[13,2,14,3],[19,10,20,11],[4,8,5,7],[1,12,2,13],[14,12,15,11],[9,18,10,19],[8,18,9,17],[5,17,6,16],[6,15,7,16]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (1,18,-2,-19)(13,4,-14,-5)(5,8,-6,-9)(15,6,-16,-7)(20,9,-1,-10)(10,19,-11,-20)(11,2,-12,-3)(3,12,-4,-13)(17,14,-18,-15)(7,16,-8,-17)
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,-19,10)(-2,11,19)(-3,-13,-5,-9,20,-11)(-4,13)(-6,15,-18,1,9)(-7,-17,-15)(-8,5,-14,17)(-10,-20)(-12,3)(-16,7)(2,18,14,4,12)(6,8,16)
Loop annotated with half-edges
12^1_94 annotated with half-edges