$12^{1}_{90}$ - Minimal pinning sets
Pinning sets for 12^1_90 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_90 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 288
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: 3.03466
on average over minimal pinning sets: 2.4
on average over optimal pinning sets: 2.4
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {2, 3, 5, 6, 9}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {1, 2, 5, 6, 9}
5
[2, 2, 2, 3, 3]
2.40
C (optimal)
• {2, 3, 5, 7, 9}
5
[2, 2, 2, 3, 3]
2.40
D (optimal)
• {1, 2, 5, 7, 9}
5
[2, 2, 2, 3, 3]
2.40
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
2.4
6
0
0
24
2.69
7
0
0
61
2.9
8
0
0
85
3.05
9
0
0
70
3.16
10
0
0
34
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
4
0
284
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,2,2],[0,1,1,3],[0,2,6,7],[0,7,5,5],[1,4,4,8],[3,8,9,9],[3,9,8,4],[5,7,9,6],[6,8,7,6]]
PD code (use to draw this loop with SnapPy): [[20,17,1,18],[18,3,19,4],[4,19,5,20],[5,16,6,17],[1,13,2,12],[2,11,3,12],[8,15,9,16],[6,14,7,13],[7,10,8,11],[14,9,15,10]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (15,20,-16,-1)(11,6,-12,-7)(4,7,-5,-8)(19,8,-20,-9)(9,18,-10,-19)(10,3,-11,-4)(5,12,-6,-13)(2,13,-3,-14)(17,14,-18,-15)(1,16,-2,-17)
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,-17,-15)(-2,-14,17)(-3,10,18,14)(-4,-8,19,-10)(-5,-13,2,16,20,8)(-6,11,3,13)(-7,4,-11)(-9,-19)(-12,5,7)(-16,1)(-18,9,-20,15)(6,12)
Loop annotated with half-edges
12^1_90 annotated with half-edges