$11^{1}_{82}$ - Minimal pinning sets
Pinning sets for 11^1_82 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^1_82 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, 2, 3, 5, 6, 10}
6
[2, 2, 2, 2, 2, 3]
2.17
B (optimal)
• {1, 2, 3, 4, 6, 10}
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, 4, 5, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,4,4,2],[0,1,4,5],[0,5,6,0],[1,6,2,1],[2,7,7,3],[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],[8,16,9,15],[17,14,18,15],[6,1,7,2],[16,10,17,9],[13,2,14,3],[5,10,6,11],[3,12,4,13],[11,4,12,5]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (8,1,-9,-2)(15,4,-16,-5)(6,13,-7,-14)(2,7,-3,-8)(18,9,-1,-10)(14,11,-15,-12)(12,5,-13,-6)(3,16,-4,-17)(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,-3,-17,10)(-2,-8)(-4,15,11,17)(-5,12,-15)(-6,-14,-12)(-7,2,-9,18,-11,14)(-10,-18)(-13,6)(-16,3,7,13,5)(1,9)(4,16)
Loop annotated with half-edges
11^1_82 annotated with half-edges