$12^{1}_{197}$ - Minimal pinning sets
Pinning sets for 12^1_197 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_197 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 216
of which optimal: 2
of which minimal: 4
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.98488
on average over minimal pinning sets: 2.35833
on average over optimal pinning sets: 2.3
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 8, 9}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 3, 6, 8, 9}
5
[2, 2, 2, 2, 4]
2.40
a (minimal)
• {1, 3, 4, 8, 9, 11}
6
[2, 2, 2, 2, 3, 4]
2.50
b (minimal)
• {1, 3, 5, 8, 9, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
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
2.3
6
0
2
13
2.58
7
0
0
43
2.81
8
0
0
64
2.98
9
0
0
55
3.11
10
0
0
28
3.2
11
0
0
8
3.27
12
0
0
1
3.33
Total
2
2
212
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: [[1,2,3,3],[0,4,4,5],[0,6,7,8],[0,8,8,0],[1,7,9,1],[1,9,6,6],[2,5,5,9],[2,9,4,8],[2,7,3,3],[4,7,6,5]]
PD code (use to draw this loop with SnapPy): [[20,15,1,16],[16,9,17,10],[12,19,13,20],[14,1,15,2],[8,17,9,18],[10,5,11,6],[6,11,7,12],[18,3,19,4],[13,3,14,2],[4,7,5,8]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (15,20,-16,-1)(7,2,-8,-3)(1,4,-2,-5)(13,6,-14,-7)(3,8,-4,-9)(17,12,-18,-13)(5,14,-6,-15)(9,16,-10,-17)(11,18,-12,-19)(19,10,-20,-11)
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,-5,-15)(-2,7,-14,5)(-3,-9,-17,-13,-7)(-4,1,-16,9)(-6,13,-18,11,-20,15)(-8,3)(-10,19,-12,17)(-11,-19)(2,4,8)(6,14)(10,16,20)(12,18)
Loop annotated with half-edges
12^1_197 annotated with half-edges