$12^{1}_{93}$ - Minimal pinning sets
Pinning sets for 12^1_93 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_93 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 240
of which optimal: 4
of which minimal: 4
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.97778
on average over minimal pinning sets: 2.3
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, 4, 5, 11}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 2, 5, 8, 11}
5
[2, 2, 2, 2, 4]
2.40
C (optimal)
• {1, 2, 5, 7, 11}
5
[2, 2, 2, 2, 4]
2.40
D (optimal)
• {1, 2, 5, 11, 12}
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
4
0
0
2.3
6
0
0
22
2.62
7
0
0
52
2.84
8
0
0
69
3.0
9
0
0
56
3.11
10
0
0
28
3.2
11
0
0
8
3.27
12
0
0
1
3.33
Total
4
0
236
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: [[1,2,3,4],[0,4,4,5],[0,5,6,6],[0,6,7,8],[0,5,1,1],[1,4,8,2],[2,7,3,2],[3,6,9,9],[3,9,9,5],[7,8,8,7]]
PD code (use to draw this loop with SnapPy): [[3,20,4,1],[2,9,3,10],[12,19,13,20],[4,18,5,17],[1,11,2,10],[11,8,12,9],[18,13,19,14],[5,14,6,15],[7,16,8,17],[6,16,7,15]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (9,6,-10,-7)(20,7,-1,-8)(8,19,-9,-20)(16,11,-17,-12)(3,12,-4,-13)(13,4,-14,-5)(5,14,-6,-15)(15,2,-16,-3)(10,17,-11,-18)(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)(-2,15,-6,9,19)(-3,-13,-5,-15)(-4,13)(-7,20,-9)(-8,-20)(-10,-18,1,7)(-11,16,2,18)(-12,3,-16)(-14,5)(-17,10,6,14,4,12)(11,17)
Loop annotated with half-edges
12^1_93 annotated with half-edges