$11^{1}_{49}$ - Minimal pinning sets
Pinning sets for 11^1_49 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^1_49 Pinning data
Pinning number of this loop: 4
Total number of pinning sets: 272
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: 3.0499
on average over minimal pinning sets: 2.575
on average over optimal pinning sets: 2.5
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 5}
4
[2, 2, 3, 3]
2.50
B (optimal)
• {1, 3, 4, 10}
4
[2, 2, 3, 3]
2.50
C (optimal)
• {1, 3, 5, 10}
4
[2, 2, 3, 3]
2.50
a (minimal)
• {1, 2, 3, 4, 11}
5
[2, 2, 3, 3, 4]
2.80
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
3
0
0
2.5
5
0
1
19
2.76
6
0
0
55
2.93
7
0
0
81
3.05
8
0
0
69
3.14
9
0
0
34
3.2
10
0
0
9
3.24
11
0
0
1
3.27
Total
3
1
268
Other information about this loop Properties
Region degree sequence: [2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,2],[0,1,5,6],[0,6,7,8],[0,8,6,5],[1,4,2,1],[2,4,7,3],[3,6,8,8],[3,7,7,4]]
PD code (use to draw this loop with SnapPy): [[18,7,1,8],[8,16,9,15],[17,14,18,15],[3,6,4,7],[1,11,2,10],[16,10,17,9],[2,13,3,14],[5,12,6,13],[4,12,5,11]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (18,3,-1,-4)(15,4,-16,-5)(6,13,-7,-14)(7,16,-8,-17)(8,1,-9,-2)(2,9,-3,-10)(17,10,-18,-11)(14,11,-15,-12)(12,5,-13,-6)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,8,16,4)(-2,-10,17,-8)(-3,18,10)(-4,15,11,-18)(-5,12,-15)(-6,-14,-12)(-7,-17,-11,14)(-9,2)(-13,6)(-16,7,13,5)(1,3,9)
Loop annotated with half-edges
11^1_49 annotated with half-edges