$12^{1}_{87}$ - Minimal pinning sets
Pinning sets for 12^1_87 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_87 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)
• {2, 3, 6, 7, 11}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {1, 2, 6, 7, 11}
5
[2, 2, 2, 3, 3]
2.40
C (optimal)
• {2, 3, 7, 8, 11}
5
[2, 2, 2, 3, 3]
2.40
D (optimal)
• {1, 2, 7, 8, 11}
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, 4, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,4,5,2],[0,1,5,0],[0,6,4,4],[1,3,3,7],[1,8,6,2],[3,5,9,7],[4,6,9,8],[5,7,9,9],[6,8,8,7]]
PD code (use to draw this loop with SnapPy): [[20,5,1,6],[6,3,7,4],[4,19,5,20],[1,15,2,14],[2,13,3,14],[7,18,8,19],[15,8,16,9],[9,12,10,13],[10,17,11,18],[16,11,17,12]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (4,1,-5,-2)(18,3,-19,-4)(13,6,-14,-7)(14,9,-15,-10)(7,10,-8,-11)(20,11,-1,-12)(12,19,-13,-20)(8,15,-9,-16)(5,16,-6,-17)(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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,4,-19,12)(-2,-18,-4)(-3,18)(-5,-17,2)(-6,13,19,3,17)(-7,-11,20,-13)(-8,-16,5,1,11)(-9,14,6,16)(-10,7,-14)(-12,-20)(-15,8,10)(9,15)
Loop annotated with half-edges
12^1_87 annotated with half-edges