$11^{1}_{57}$ - Minimal pinning sets
Pinning sets for 11^1_57 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^1_57 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 192
of which optimal: 6
of which minimal: 6
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.04438
on average over minimal pinning sets: 2.6
on average over optimal pinning sets: 2.6
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 4, 6, 8}
5
[2, 2, 3, 3, 3]
2.60
B (optimal)
• {1, 3, 4, 6, 7}
5
[2, 2, 3, 3, 3]
2.60
C (optimal)
• {1, 3, 6, 8, 11}
5
[2, 2, 3, 3, 3]
2.60
D (optimal)
• {1, 3, 6, 7, 11}
5
[2, 2, 3, 3, 3]
2.60
E (optimal)
• {1, 2, 4, 6, 8}
5
[2, 2, 3, 3, 3]
2.60
F (optimal)
• {1, 2, 4, 6, 7}
5
[2, 2, 3, 3, 3]
2.60
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
6
0
0
2.6
6
0
0
29
2.84
7
0
0
57
3.0
8
0
0
58
3.12
9
0
0
32
3.19
10
0
0
9
3.24
11
0
0
1
3.27
Total
6
0
186
Other information about this loop Properties
Region degree sequence: [2, 2, 3, 3, 3, 3, 3, 3, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,2,0],[0,1,5,3],[0,2,6,4],[1,3,6,7],[2,7,8,8],[3,8,7,4],[4,6,8,5],[5,7,6,5]]
PD code (use to draw this loop with SnapPy): [[11,18,12,1],[17,10,18,11],[12,10,13,9],[1,9,2,8],[16,7,17,8],[13,4,14,5],[2,15,3,16],[3,6,4,7],[14,6,15,5]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (11,2,-12,-3)(3,12,-4,-13)(1,4,-2,-5)(14,5,-15,-6)(6,17,-7,-18)(7,10,-8,-11)(15,8,-16,-9)(18,13,-1,-14)(9,16,-10,-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,-5,14)(-2,11,-8,15,5)(-3,-13,18,-7,-11)(-4,1,13)(-6,-18,-14)(-9,-17,6,-15)(-10,7,17)(-12,3)(-16,9)(2,4,12)(8,10,16)
Loop annotated with half-edges
11^1_57 annotated with half-edges