$11^{1}_{99}$ - Minimal pinning sets
Pinning sets for 11^1_99 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^1_99 Pinning data
Pinning number of this loop: 4
Total number of pinning sets: 144
of which optimal: 1
of which minimal: 2
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.97128
on average over minimal pinning sets: 2.375
on average over optimal pinning sets: 2.25
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 6, 10}
4
[2, 2, 2, 3]
2.25
a (minimal)
• {1, 3, 4, 7, 8, 10}
6
[2, 2, 2, 3, 3, 3]
2.50
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
1
0
0
2.25
5
0
0
7
2.57
6
0
1
21
2.77
7
0
0
39
2.92
8
0
0
41
3.05
9
0
0
25
3.15
10
0
0
8
3.23
11
0
0
1
3.27
Total
1
1
142
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 3, 5, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,2],[0,1,5,3],[0,2,6,6],[0,7,7,1],[1,8,8,2],[3,8,7,3],[4,6,8,4],[5,7,6,5]]
PD code (use to draw this loop with SnapPy): [[18,5,1,6],[6,11,7,12],[12,17,13,18],[13,4,14,5],[1,10,2,11],[7,16,8,17],[3,14,4,15],[9,2,10,3],[15,8,16,9]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (7,18,-8,-1)(11,2,-12,-3)(16,3,-17,-4)(13,6,-14,-7)(17,8,-18,-9)(4,9,-5,-10)(1,12,-2,-13)(5,14,-6,-15)(10,15,-11,-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)(-17,17)(-18,18)
Face permutation $\varphi=(\sigma\epsilon)^{-1}=$ (-1,-13,-7)(-2,11,15,-6,13)(-3,16,-11)(-4,-10,-16)(-5,-15,10)(-8,17,3,-12,1)(-9,4,-17)(-14,5,9,-18,7)(2,12)(6,14)(8,18)
Loop annotated with half-edges
11^1_99 annotated with half-edges