$12^{1}_{238}$ - Minimal pinning sets
Pinning sets for 12^1_238 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_238 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 224
of which optimal: 3
of which minimal: 3
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.9785
on average over minimal pinning sets: 2.26667
on average over optimal pinning sets: 2.26667
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 4, 7, 9, 11}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 3, 4, 9, 11}
5
[2, 2, 2, 2, 3]
2.20
C (optimal)
• {1, 4, 6, 9, 11}
5
[2, 2, 2, 2, 4]
2.40
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
2.27
6
0
0
18
2.59
7
0
0
46
2.82
8
0
0
65
2.98
9
0
0
55
3.11
10
0
0
28
3.2
11
0
0
8
3.27
12
0
0
1
3.33
Total
3
0
221
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,6,7],[0,7,3,3],[0,2,2,8],[0,8,9,9],[1,9,9,8],[1,8,7,7],[1,6,6,2],[3,6,5,4],[4,5,5,4]]
PD code (use to draw this loop with SnapPy): [[17,20,18,1],[13,16,14,17],[19,6,20,7],[18,6,19,5],[1,10,2,11],[3,12,4,13],[15,8,16,9],[14,8,15,7],[9,4,10,5],[2,12,3,11]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (6,1,-7,-2)(2,15,-3,-16)(16,3,-17,-4)(4,11,-5,-12)(12,5,-13,-6)(18,7,-19,-8)(8,19,-9,-20)(20,9,-1,-10)(10,13,-11,-14)(14,17,-15,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,6,-13,10)(-2,-16,-4,-12,-6)(-3,16)(-5,12)(-7,18,-15,2)(-8,-20,-10,-14,-18)(-9,20)(-11,4,-17,14)(-19,8)(1,9,19,7)(3,15,17)(5,11,13)
Loop annotated with half-edges
12^1_238 annotated with half-edges