$12^{1}_{102}$ - Minimal pinning sets
Pinning sets for 12^1_102 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_102 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, 3, 4, 7, 11}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 4, 5, 7, 11}
5
[2, 2, 2, 2, 4]
2.40
C (optimal)
• {1, 4, 6, 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
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: [[1,2,2,3],[0,3,3,4],[0,5,6,0],[0,4,1,1],[1,3,7,5],[2,4,8,6],[2,5,9,9],[4,9,8,8],[5,7,7,9],[6,8,7,6]]
PD code (use to draw this loop with SnapPy): [[3,20,4,1],[2,9,3,10],[19,4,20,5],[1,11,2,10],[11,8,12,9],[5,12,6,13],[13,18,14,19],[7,16,8,17],[6,16,7,15],[17,14,18,15]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (17,2,-18,-3)(9,6,-10,-7)(20,7,-1,-8)(8,19,-9,-20)(3,10,-4,-11)(15,12,-16,-13)(13,4,-14,-5)(5,14,-6,-15)(11,16,-12,-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,-19,8)(-2,17,-12,15,-6,9,19)(-3,-11,-17)(-4,13,-16,11)(-5,-15,-13)(-7,20,-9)(-8,-20)(-10,3,-18,1,7)(-14,5)(2,18)(4,10,6,14)(12,16)
Loop annotated with half-edges
12^1_102 annotated with half-edges