$12^{1}_{173}$ - Minimal pinning sets
Pinning sets for 12^1_173 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_173 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 272
of which optimal: 3
of which minimal: 4
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.0386
on average over minimal pinning sets: 2.46667
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, 3, 4, 8, 11}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {1, 2, 4, 8, 11}
5
[2, 2, 2, 3, 3]
2.40
C (optimal)
• {1, 3, 4, 6, 11}
5
[2, 2, 2, 3, 3]
2.40
a (minimal)
• {1, 2, 4, 6, 7, 11}
6
[2, 2, 2, 3, 3, 4]
2.67
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
1
19
2.68
7
0
0
55
2.89
8
0
0
81
3.04
9
0
0
69
3.15
10
0
0
34
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
3
1
268
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: [[1,2,3,4],[0,5,5,2],[0,1,5,6],[0,6,7,8],[0,8,9,9],[1,9,2,1],[2,9,7,3],[3,6,8,8],[3,7,7,4],[4,6,5,4]]
PD code (use to draw this loop with SnapPy): [[5,20,6,1],[4,17,5,18],[19,16,20,17],[6,10,7,9],[1,13,2,12],[18,3,19,4],[15,10,16,11],[7,15,8,14],[8,13,9,14],[2,11,3,12]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (16,1,-17,-2)(7,4,-8,-5)(13,8,-14,-9)(20,9,-1,-10)(10,19,-11,-20)(6,11,-7,-12)(12,5,-13,-6)(17,14,-18,-15)(2,15,-3,-16)(3,18,-4,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,16,-3,-19,10)(-2,-16)(-4,7,11,19)(-5,12,-7)(-6,-12)(-8,13,5)(-9,20,-11,6,-13)(-10,-20)(-14,17,1,9)(-15,2,-17)(-18,3,15)(4,18,14,8)
Loop annotated with half-edges
12^1_173 annotated with half-edges