$12^{1}_{108}$ - Minimal pinning sets
Pinning sets for 12^1_108 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_108 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: 3.04277
on average over minimal pinning sets: 2.45
on average over optimal pinning sets: 2.45
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, 7, 11}
5
[2, 2, 2, 3, 3]
2.40
C (optimal)
• {1, 2, 5, 11, 12}
5
[2, 2, 2, 3, 3]
2.40
D (optimal)
• {1, 2, 5, 8, 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
4
0
0
2.45
6
0
0
24
2.72
7
0
0
61
2.91
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, 4, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,4,5],[0,5,6,3],[0,2,6,7],[0,5,1,1],[1,4,8,2],[2,8,7,3],[3,6,9,9],[5,9,9,6],[7,8,8,7]]
PD code (use to draw this loop with SnapPy): [[3,20,4,1],[13,2,14,3],[19,10,20,11],[4,10,5,9],[1,12,2,13],[14,12,15,11],[18,5,19,6],[8,17,9,18],[15,7,16,6],[16,7,17,8]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (11,20,-12,-1)(16,3,-17,-4)(4,13,-5,-14)(14,5,-15,-6)(7,2,-8,-3)(17,8,-18,-9)(9,12,-10,-13)(19,10,-20,-11)(6,15,-7,-16)(1,18,-2,-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,-19,-11)(-2,7,15,5,13,-10,19)(-3,16,-7)(-4,-14,-6,-16)(-5,14)(-8,17,3)(-9,-13,4,-17)(-12,9,-18,1)(-15,6)(-20,11)(2,18,8)(10,12,20)
Loop annotated with half-edges
12^1_108 annotated with half-edges