$12^{1}_{148}$ - Minimal pinning sets
Pinning sets for 12^1_148 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_148 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)
• {2, 3, 4, 7, 11}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 2, 4, 7, 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: [[1,2,3,3],[0,3,4,5],[0,6,6,7],[0,8,1,0],[1,8,9,5],[1,4,9,9],[2,9,7,2],[2,6,8,8],[3,7,7,4],[4,6,5,5]]
PD code (use to draw this loop with SnapPy): [[13,20,14,1],[15,12,16,13],[19,4,20,5],[14,2,15,1],[8,11,9,12],[16,9,17,10],[5,18,6,19],[6,3,7,4],[2,7,3,8],[10,17,11,18]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (3,20,-4,-1)(16,1,-17,-2)(19,4,-20,-5)(13,6,-14,-7)(5,8,-6,-9)(12,9,-13,-10)(7,14,-8,-15)(2,15,-3,-16)(10,17,-11,-18)(18,11,-19,-12)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,16,-3)(-2,-16)(-4,19,11,17,1)(-5,-9,12,-19)(-6,13,9)(-7,-15,2,-17,10,-13)(-8,5,-20,3,15)(-10,-18,-12)(-11,18)(-14,7)(4,20)(6,8,14)
Loop annotated with half-edges
12^1_148 annotated with half-edges