$12^{1}_{339}$ - Minimal pinning sets
Pinning sets for 12^1_339 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_339 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 336
of which optimal: 6
of which minimal: 6
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.04915
on average over minimal pinning sets: 2.5
on average over optimal pinning sets: 2.5
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 8, 10, 11}
5
[2, 2, 2, 3, 4]
2.60
B (optimal)
• {1, 3, 6, 10, 11}
5
[2, 2, 2, 3, 3]
2.40
C (optimal)
• {1, 3, 7, 10, 11}
5
[2, 2, 2, 3, 3]
2.40
D (optimal)
• {1, 3, 7, 9, 11}
5
[2, 2, 2, 3, 4]
2.60
E (optimal)
• {1, 2, 3, 7, 11}
5
[2, 2, 2, 3, 3]
2.40
F (optimal)
• {1, 3, 5, 10, 11}
5
[2, 2, 2, 3, 4]
2.60
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
6
0
0
2.5
6
0
0
33
2.76
7
0
0
77
2.94
8
0
0
99
3.07
9
0
0
76
3.17
10
0
0
35
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
6
0
330
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,6,7],[0,7,8,3],[0,2,9,4],[0,3,5,5],[1,4,4,6],[1,5,9,9],[1,8,8,2],[2,7,7,9],[3,8,6,6]]
PD code (use to draw this loop with SnapPy): [[9,20,10,1],[3,8,4,9],[19,16,20,17],[10,16,11,15],[1,15,2,14],[2,13,3,14],[7,12,8,13],[4,18,5,17],[5,18,6,19],[11,6,12,7]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (9,4,-10,-5)(1,6,-2,-7)(5,10,-6,-11)(20,11,-1,-12)(12,19,-13,-20)(13,8,-14,-9)(17,14,-18,-15)(15,2,-16,-3)(3,16,-4,-17)(7,18,-8,-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,-7,-19,12)(-2,15,-18,7)(-3,-17,-15)(-4,9,-14,17)(-5,-11,20,-13,-9)(-6,1,11)(-8,13,19)(-10,5)(-12,-20)(-16,3)(2,6,10,4,16)(8,18,14)
Loop annotated with half-edges
12^1_339 annotated with half-edges