$10^{1}_{22}$ - Minimal pinning sets
Pinning sets for 10^1_22 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 10^1_22 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 60
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: 2.83683
on average over minimal pinning sets: 2.3
on average over optimal pinning sets: 2.3
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 5, 8, 9}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 2, 5, 7, 9}
5
[2, 2, 2, 2, 4]
2.40
C (optimal)
• {1, 2, 4, 5, 9}
5
[2, 2, 2, 2, 4]
2.40
D (optimal)
• {1, 2, 3, 5, 9}
5
[2, 2, 2, 2, 3]
2.20
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
2.3
6
0
0
14
2.64
7
0
0
20
2.86
8
0
0
15
3.0
9
0
0
6
3.11
10
0
0
1
3.2
Total
4
0
56
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,4,5,5],[0,6,6,3],[0,2,7,0],[1,7,7,5],[1,4,6,1],[2,5,7,2],[3,6,4,4]]
PD code (use to draw this loop with SnapPy): [[16,11,1,12],[12,8,13,7],[15,2,16,3],[10,1,11,2],[8,5,9,6],[13,6,14,7],[3,14,4,15],[4,9,5,10]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (5,2,-6,-3)(10,3,-11,-4)(4,9,-5,-10)(1,6,-2,-7)(13,8,-14,-9)(16,11,-1,-12)(7,14,-8,-15)(12,15,-13,-16)
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)
Face permutation $\varphi=(\sigma\epsilon)^{-1}=$ (-1,-7,-15,12)(-2,5,9,-14,7)(-3,10,-5)(-4,-10)(-6,1,11,3)(-8,13,15)(-9,4,-11,16,-13)(-12,-16)(2,6)(8,14)
Loop annotated with half-edges
10^1_22 annotated with half-edges