$12^{1}_{103}$ - Minimal pinning sets
Pinning sets for 12^1_103 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_103 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 224
of which optimal: 3
of which minimal: 3
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.9785
on average over minimal pinning sets: 2.26667
on average over optimal pinning sets: 2.26667
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 7, 11}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 2, 3, 5, 11}
5
[2, 2, 2, 2, 4]
2.40
C (optimal)
• {1, 2, 3, 4, 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
3
0
0
2.27
6
0
0
18
2.59
7
0
0
46
2.82
8
0
0
65
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
3
0
221
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 5, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,4,5],[0,6,7,3],[0,2,8,8],[0,5,1,1],[1,4,6,6],[2,5,5,9],[2,9,9,8],[3,7,9,3],[6,8,7,7]]
PD code (use to draw this loop with SnapPy): [[3,20,4,1],[2,13,3,14],[19,10,20,11],[4,10,5,9],[1,15,2,14],[15,12,16,13],[11,16,12,17],[7,18,8,19],[5,8,6,9],[17,6,18,7]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (15,4,-16,-5)(3,6,-4,-7)(7,2,-8,-3)(17,8,-18,-9)(13,10,-14,-11)(20,11,-1,-12)(12,19,-13,-20)(9,14,-10,-15)(5,16,-6,-17)(1,18,-2,-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,-19,12)(-2,7,-4,15,-10,13,19)(-3,-7)(-5,-17,-9,-15)(-6,3,-8,17)(-11,20,-13)(-12,-20)(-14,9,-18,1,11)(-16,5)(2,18,8)(4,6,16)(10,14)
Loop annotated with half-edges
12^1_103 annotated with half-edges