$12^{1}_{88}$ - Minimal pinning sets
Pinning sets for 12^1_88 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_88 Pinning data
Pinning number of this loop: 4
Total number of pinning sets: 352
of which optimal: 1
of which minimal: 3
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.04132
on average over minimal pinning sets: 2.41667
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, 4, 8, 9}
4
[2, 2, 2, 3]
2.25
a (minimal)
• {1, 4, 5, 9, 10}
5
[2, 2, 2, 3, 3]
2.40
b (minimal)
• {1, 4, 6, 9, 10}
5
[2, 2, 2, 3, 4]
2.60
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
2
8
2.56
6
0
0
39
2.79
7
0
0
81
2.95
8
0
0
100
3.08
9
0
0
76
3.17
10
0
0
35
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
1
2
349
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: [[1,1,2,3],[0,4,2,0],[0,1,5,6],[0,7,7,4],[1,3,7,8],[2,8,9,9],[2,9,8,7],[3,6,4,3],[4,6,9,5],[5,8,6,5]]
PD code (use to draw this loop with SnapPy): [[20,7,1,8],[8,19,9,20],[9,6,10,7],[1,17,2,16],[18,15,19,16],[12,5,13,6],[10,4,11,3],[17,3,18,2],[11,14,12,15],[4,13,5,14]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (3,20,-4,-1)(14,1,-15,-2)(18,7,-19,-8)(5,8,-6,-9)(12,9,-13,-10)(10,15,-11,-16)(2,13,-3,-14)(16,11,-17,-12)(17,4,-18,-5)(6,19,-7,-20)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,14,-3)(-2,-14)(-4,17,11,15,1)(-5,-9,12,-17)(-6,-20,3,13,9)(-7,18,4,20)(-8,5,-18)(-10,-16,-12)(-11,16)(-13,2,-15,10)(-19,6,8)(7,19)
Loop annotated with half-edges
12^1_88 annotated with half-edges