$12^{1}_{203}$ - Minimal pinning sets
Pinning sets for 12^1_203 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_203 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 284
of which optimal: 3
of which minimal: 6
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.03793
on average over minimal pinning sets: 2.53333
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)
• {1, 2, 5, 9, 10}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {1, 2, 3, 9, 11}
5
[2, 2, 2, 3, 3]
2.40
C (optimal)
• {1, 2, 5, 9, 11}
5
[2, 2, 2, 3, 3]
2.40
a (minimal)
• {1, 2, 3, 6, 9, 10}
6
[2, 2, 2, 3, 3, 3]
2.50
b (minimal)
• {1, 2, 3, 7, 9, 10}
6
[2, 2, 2, 3, 3, 4]
2.67
c (minimal)
• {1, 2, 3, 4, 9, 10}
6
[2, 2, 2, 3, 3, 5]
2.83
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
3
0
0
2.4
6
0
3
19
2.68
7
0
0
60
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
3
3
278
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 3, 4, 5, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,2],[0,1,6,7],[0,7,7,6],[0,8,9,9],[1,9,8,1],[2,8,3,7],[2,6,3,3],[4,6,5,9],[4,8,5,4]]
PD code (use to draw this loop with SnapPy): [[5,20,6,1],[4,9,5,10],[19,8,20,9],[6,17,7,18],[1,12,2,13],[10,3,11,4],[15,18,16,19],[16,7,17,8],[11,14,12,15],[2,14,3,13]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (7,20,-8,-1)(13,4,-14,-5)(16,5,-17,-6)(1,6,-2,-7)(18,9,-19,-10)(10,19,-11,-20)(8,11,-9,-12)(3,14,-4,-15)(12,15,-13,-16)(2,17,-3,-18)
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)(-2,-18,-10,-20,7)(-3,-15,12,-9,18)(-4,13,15)(-5,16,-13)(-6,1,-8,-12,-16)(-11,8,20)(-14,3,17,5)(-17,2,6)(-19,10)(4,14)(9,11,19)
Loop annotated with half-edges
12^1_203 annotated with half-edges