$12^{1}_{130}$ - Minimal pinning sets
Pinning sets for 12^1_130 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_130 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, 6, 7}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 2, 5, 6, 7}
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, 4, 4, 4, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,4,0],[0,4,5,6],[0,7,8,4],[1,3,2,1],[2,9,9,6],[2,5,9,7],[3,6,8,8],[3,7,7,9],[5,8,6,5]]
PD code (use to draw this loop with SnapPy): [[7,20,8,1],[19,6,20,7],[8,18,9,17],[1,4,2,5],[5,18,6,19],[9,15,10,14],[16,13,17,14],[3,12,4,13],[2,12,3,11],[15,11,16,10]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (19,2,-20,-3)(17,4,-18,-5)(8,13,-9,-14)(9,20,-10,-1)(1,10,-2,-11)(14,11,-15,-12)(12,7,-13,-8)(15,6,-16,-7)(3,16,-4,-17)(5,18,-6,-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,-11,14,-9)(-2,19,-6,15,11)(-3,-17,-5,-19)(-4,17)(-7,12,-15)(-8,-14,-12)(-10,1)(-13,8)(-16,3,-20,9,13,7)(-18,5)(2,10,20)(4,16,6,18)
Loop annotated with half-edges
12^1_130 annotated with half-edges