$11^{1}_{87}$ - Minimal pinning sets
Pinning sets for 11^1_87 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^1_87 Pinning data
Pinning number of this loop: 6
Total number of pinning sets: 48
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: 2.84761
on average over minimal pinning sets: 2.16667
on average over optimal pinning sets: 2.16667
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 5, 6, 7, 8}
6
[2, 2, 2, 2, 2, 3]
2.17
B (optimal)
• {1, 3, 4, 6, 7, 8}
6
[2, 2, 2, 2, 2, 3]
2.17
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
6
2
0
0
2.17
7
0
0
9
2.54
8
0
0
16
2.81
9
0
0
14
3.02
10
0
0
6
3.17
11
0
0
1
3.27
Total
2
0
46
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 2, 3, 3, 3, 4, 6, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,2,0],[0,1,5,3],[0,2,6,6],[1,7,5,5],[2,4,4,7],[3,8,8,3],[4,8,8,5],[6,7,7,6]]
PD code (use to draw this loop with SnapPy): [[13,18,14,1],[17,12,18,13],[14,12,15,11],[1,11,2,10],[5,16,6,17],[15,6,16,7],[2,9,3,10],[4,7,5,8],[8,3,9,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (10,1,-11,-2)(17,2,-18,-3)(15,4,-16,-5)(5,14,-6,-15)(6,9,-7,-10)(12,7,-13,-8)(18,11,-1,-12)(8,13,-9,-14)(3,16,-4,-17)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,10,-7,12)(-2,17,-4,15,-6,-10)(-3,-17)(-5,-15)(-8,-14,5,-16,3,-18,-12)(-9,6,14)(-11,18,2)(-13,8)(1,11)(4,16)(7,9,13)
Loop annotated with half-edges
11^1_87 annotated with half-edges