$11^{1}_{45}$ - Minimal pinning sets
Pinning sets for 11^1_45 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^1_45 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 96
of which optimal: 2
of which minimal: 2
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.90403
on average over minimal pinning sets: 2.2
on average over optimal pinning sets: 2.2
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 5, 8, 10}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 3, 4, 8, 10}
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
2
0
0
2.2
6
0
0
11
2.55
7
0
0
25
2.79
8
0
0
30
2.97
9
0
0
20
3.1
10
0
0
7
3.2
11
0
0
1
3.27
Total
2
0
94
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 4, 4, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,2,0],[0,1,5,3],[0,2,6,4],[1,3,6,5],[2,4,7,7],[3,8,8,4],[5,8,8,5],[6,7,7,6]]
PD code (use to draw this loop with SnapPy): [[18,7,1,8],[8,17,9,18],[9,6,10,7],[1,10,2,11],[11,16,12,17],[12,5,13,6],[2,15,3,16],[4,13,5,14],[14,3,15,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (8,1,-9,-2)(5,2,-6,-3)(14,3,-15,-4)(15,6,-16,-7)(18,9,-1,-10)(16,11,-17,-12)(7,12,-8,-13)(4,13,-5,-14)(10,17,-11,-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)
Face permutation $\varphi=(\sigma\epsilon)^{-1}=$ (-1,8,12,-17,10)(-2,5,13,-8)(-3,14,-5)(-4,-14)(-6,15,3)(-7,-13,4,-15)(-9,18,-11,16,6,2)(-10,-18)(-12,7,-16)(1,9)(11,17)
Loop annotated with half-edges
11^1_45 annotated with half-edges