$12^{1}_{220}$ - Minimal pinning sets
Pinning sets for 12^1_220 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_220 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, 4, 7, 9}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 2, 4, 7, 8}
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, 5, 5, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,2,0],[0,1,5,3],[0,2,6,7],[1,7,7,8],[2,8,8,9],[3,9,9,7],[3,6,4,4],[4,9,5,5],[5,8,6,6]]
PD code (use to draw this loop with SnapPy): [[20,13,1,14],[14,19,15,20],[15,12,16,13],[1,16,2,17],[9,18,10,19],[11,4,12,5],[2,7,3,8],[17,8,18,9],[10,6,11,5],[6,3,7,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (4,1,-5,-2)(9,2,-10,-3)(3,8,-4,-9)(20,5,-1,-6)(15,6,-16,-7)(13,10,-14,-11)(18,11,-19,-12)(19,14,-20,-15)(7,16,-8,-17)(12,17,-13,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,4,8,16,6)(-2,9,-4)(-3,-9)(-5,20,14,10,2)(-6,15,-20)(-7,-17,12,-19,-15)(-8,3,-10,13,17)(-11,18,-13)(-12,-18)(-14,19,11)(-16,7)(1,5)
Loop annotated with half-edges
12^1_220 annotated with half-edges