$12^{1}_{61}$ - Minimal pinning sets
Pinning sets for 12^1_61 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_61 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 312
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: 3.04121
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, 2, 5, 7, 11}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {1, 2, 3, 5, 11}
5
[2, 2, 2, 3, 3]
2.40
C (optimal)
• {1, 2, 3, 4, 11}
5
[2, 2, 2, 3, 3]
2.40
D (optimal)
• {1, 2, 5, 8, 11}
5
[2, 2, 2, 3, 5]
2.80
E (optimal)
• {1, 2, 5, 10, 11}
5
[2, 2, 2, 3, 3]
2.40
F (optimal)
• {1, 2, 5, 9, 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
31
2.76
7
0
0
70
2.93
8
0
0
90
3.06
9
0
0
71
3.16
10
0
0
34
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
6
0
306
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,6],[0,7,5,3],[0,2,8,8],[0,8,8,1],[1,2,9,6],[1,5,9,7],[2,6,9,9],[3,4,4,3],[5,7,7,6]]
PD code (use to draw this loop with SnapPy): [[9,20,10,1],[13,8,14,9],[19,4,20,5],[10,4,11,3],[1,12,2,13],[7,18,8,19],[14,18,15,17],[5,17,6,16],[11,2,12,3],[6,15,7,16]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (9,20,-10,-1)(1,8,-2,-9)(17,2,-18,-3)(11,6,-12,-7)(19,10,-20,-11)(4,13,-5,-14)(14,3,-15,-4)(15,12,-16,-13)(5,16,-6,-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,-9)(-2,17,-6,11,-20,9)(-3,14,-5,-17)(-4,-14)(-7,-19,-11)(-8,1,-10,19)(-12,15,3,-18,7)(-13,4,-15)(-16,5,13)(2,8,18)(6,16,12)(10,20)
Loop annotated with half-edges
12^1_61 annotated with half-edges