$11^{1}_{73}$ - Minimal pinning sets
Pinning sets for 11^1_73 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^1_73 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 88
of which optimal: 1
of which minimal: 3
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.91292
on average over minimal pinning sets: 2.34444
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, 2, 5, 6, 8}
5
[2, 2, 2, 2, 3]
2.20
a (minimal)
• {1, 2, 4, 6, 8, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
b (minimal)
• {1, 2, 4, 6, 7, 8}
6
[2, 2, 2, 2, 3, 4]
2.50
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
1
0
0
2.2
6
0
2
6
2.5
7
0
0
22
2.76
8
0
0
29
2.96
9
0
0
20
3.1
10
0
0
7
3.2
11
0
0
1
3.27
Total
1
2
85
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: [[1,2,3,3],[0,4,5,6],[0,6,6,3],[0,2,7,0],[1,8,8,5],[1,4,7,6],[1,5,2,2],[3,5,8,8],[4,7,7,4]]
PD code (use to draw this loop with SnapPy): [[18,9,1,10],[10,15,11,16],[17,2,18,3],[8,1,9,2],[14,5,15,6],[11,5,12,4],[16,4,17,3],[12,7,13,8],[6,13,7,14]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (15,18,-16,-1)(1,14,-2,-15)(2,9,-3,-10)(12,3,-13,-4)(10,5,-11,-6)(16,7,-17,-8)(4,11,-5,-12)(8,13,-9,-14)(6,17,-7,-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,-15)(-2,-10,-6,-18,15)(-3,12,-5,10)(-4,-12)(-7,16,18)(-8,-14,1,-16)(-9,2,14)(-11,4,-13,8,-17,6)(3,9,13)(5,11)(7,17)
Loop annotated with half-edges
11^1_73 annotated with half-edges