$11^{1}_{51}$ - Minimal pinning sets
Pinning sets for 11^1_51 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^1_51 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 136
of which optimal: 3
of which minimal: 4
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.9775
on average over minimal pinning sets: 2.46667
on average over optimal pinning sets: 2.4
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 5, 8, 9}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {1, 4, 5, 8, 10}
5
[2, 2, 2, 3, 3]
2.40
C (optimal)
• {1, 3, 5, 8, 10}
5
[2, 2, 2, 3, 3]
2.40
a (minimal)
• {1, 2, 4, 5, 8, 9}
6
[2, 2, 2, 3, 3, 4]
2.67
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
3
0
0
2.4
6
0
1
16
2.69
7
0
0
38
2.89
8
0
0
43
3.05
9
0
0
26
3.15
10
0
0
8
3.23
11
0
0
1
3.27
Total
3
1
132
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,4,4,2],[0,1,4,5],[0,6,7,0],[1,7,2,1],[2,7,8,6],[3,5,8,8],[3,8,5,4],[5,7,6,6]]
PD code (use to draw this loop with SnapPy): [[7,18,8,1],[6,15,7,16],[17,14,18,15],[8,2,9,1],[16,5,17,6],[10,13,11,14],[2,11,3,12],[9,4,10,5],[12,3,13,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (5,18,-6,-1)(12,1,-13,-2)(16,7,-17,-8)(6,9,-7,-10)(13,10,-14,-11)(2,11,-3,-12)(3,14,-4,-15)(15,4,-16,-5)(8,17,-9,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,12,-3,-15,-5)(-2,-12)(-4,15)(-6,-10,13,1)(-7,16,4,14,10)(-8,-18,5,-16)(-9,6,18)(-11,2,-13)(-14,3,11)(-17,8)(7,9,17)
Loop annotated with half-edges
11^1_51 annotated with half-edges