$12^{1}_{259}$ - Minimal pinning sets
Pinning sets for 12^1_259 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_259 Pinning data
Pinning number of this loop: 6
Total number of pinning sets: 220
of which optimal: 5
of which minimal: 9
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.09682
on average over minimal pinning sets: 2.68783
on average over optimal pinning sets: 2.66667
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {2, 3, 4, 7, 8, 11}
6
[2, 2, 3, 3, 3, 3]
2.67
B (optimal)
• {2, 3, 4, 6, 7, 11}
6
[2, 2, 3, 3, 3, 3]
2.67
C (optimal)
• {1, 2, 5, 6, 7, 11}
6
[2, 2, 3, 3, 3, 3]
2.67
D (optimal)
• {1, 5, 6, 7, 11, 12}
6
[2, 2, 3, 3, 3, 3]
2.67
E (optimal)
• {2, 3, 5, 6, 7, 11}
6
[2, 2, 3, 3, 3, 3]
2.67
a (minimal)
• {1, 3, 4, 7, 8, 11, 12}
7
[2, 2, 3, 3, 3, 3, 3]
2.71
b (minimal)
• {1, 3, 4, 6, 7, 11, 12}
7
[2, 2, 3, 3, 3, 3, 3]
2.71
c (minimal)
• {1, 2, 4, 5, 7, 8, 11}
7
[2, 2, 3, 3, 3, 3, 3]
2.71
d (minimal)
• {1, 4, 5, 7, 8, 11, 12}
7
[2, 2, 3, 3, 3, 3, 3]
2.71
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
6
5
0
0
2.67
7
0
4
26
2.86
8
0
0
66
3.03
9
0
0
70
3.17
10
0
0
38
3.26
11
0
0
10
3.31
12
0
0
1
3.33
Total
5
4
211
Other information about this loop Properties
Region degree sequence: [2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 6, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,4,5,2],[0,1,5,0],[0,6,7,7],[1,8,9,5],[1,4,9,2],[3,9,8,7],[3,6,8,3],[4,7,6,9],[4,8,6,5]]
PD code (use to draw this loop with SnapPy): [[7,20,8,1],[6,9,7,10],[19,8,20,9],[1,15,2,14],[10,17,11,18],[18,5,19,6],[15,12,16,13],[2,13,3,14],[3,16,4,17],[11,4,12,5]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (9,2,-10,-3)(16,3,-17,-4)(17,6,-18,-7)(4,7,-5,-8)(13,10,-14,-11)(20,11,-1,-12)(12,19,-13,-20)(1,14,-2,-15)(8,15,-9,-16)(5,18,-6,-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,-15,8,-5,-19,12)(-2,9,15)(-3,16,-9)(-4,-8,-16)(-6,17,3,-10,13,19)(-7,4,-17)(-11,20,-13)(-12,-20)(-14,1,11)(-18,5,7)(2,14,10)(6,18)
Loop annotated with half-edges
12^1_259 annotated with half-edges