- Minimal pinning sets
Pinning sets for 12^1_230 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_230 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)
• {1, 2, 4, 5, 11}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {1, 2, 4, 6, 11}
5
[2, 2, 2, 3, 3]
2.40
C (optimal)
• {1, 2, 4, 5, 9}
5
[2, 2, 2, 3, 3]
2.40
D (optimal)
• {1, 2, 4, 6, 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, 3, 4, 4, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,6],[0,6,7,8],[0,9,9,4],[0,3,9,5],[1,4,6,1],[1,5,7,2],[2,6,8,8],[2,7,7,9],[3,8,4,3]]
PD code (use to draw this loop with SnapPy): [[7,20,8,1],[6,13,7,14],[16,19,17,20],[8,3,9,4],[1,4,2,5],[14,5,15,6],[15,12,16,13],[18,11,19,12],[17,11,18,10],[2,9,3,10]]
Permutation representation (action on half-edges):
Vertex permutation (6,1,-7,-2)(19,2,-20,-3)(12,3,-13,-4)(20,7,-1,-8)(8,15,-9,-16)(16,9,-17,-10)(13,10,-14,-11)(4,11,-5,-12)(14,17,-15,-18)(5,18,-6,-19)
Edge permutation (-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 (-1,6,18,-15,8)(-2,19,-6)(-3,12,-5,-19)(-4,-12)(-7,20,2)(-8,-16,-10,13,3,-20)(-9,16)(-11,4,-13)(-14,-18,5,11)(-17,14,10)(1,7)(9,15,17)
Loop annotated with half-edges
12^1_230 annotated with half-edges