$11^{1}_{28}$ - Minimal pinning sets
Pinning sets for 11^1_28 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^1_28 Pinning data
Pinning number of this loop: 4
Total number of pinning sets: 248
of which optimal: 5
of which minimal: 5
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.97657
on average over minimal pinning sets: 2.4
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, 4, 6, 10}
4
[2, 2, 2, 3]
2.25
B (optimal)
• {1, 4, 5, 10}
4
[2, 2, 2, 4]
2.50
C (optimal)
• {1, 4, 10, 11}
4
[2, 2, 2, 3]
2.25
D (optimal)
• {1, 4, 9, 10}
4
[2, 2, 2, 5]
2.75
E (optimal)
• {1, 2, 4, 10}
4
[2, 2, 2, 3]
2.25
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
5
0
0
2.4
5
0
0
25
2.69
6
0
0
55
2.87
7
0
0
70
3.0
8
0
0
56
3.09
9
0
0
28
3.17
10
0
0
8
3.23
11
0
0
1
3.27
Total
5
0
243
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: [[1,2,3,4],[0,4,4,5],[0,6,6,3],[0,2,7,8],[0,5,1,1],[1,4,8,6],[2,5,7,2],[3,6,8,8],[3,7,7,5]]
PD code (use to draw this loop with SnapPy): [[3,18,4,1],[2,13,3,14],[17,8,18,9],[4,8,5,7],[1,15,2,14],[15,12,16,13],[9,16,10,17],[5,10,6,11],[11,6,12,7]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (9,4,-10,-5)(5,2,-6,-3)(15,6,-16,-7)(3,8,-4,-9)(13,10,-14,-11)(18,11,-1,-12)(12,17,-13,-18)(7,14,-8,-15)(1,16,-2,-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,-17,12)(-2,5,-10,13,17)(-3,-9,-5)(-4,9)(-6,15,-8,3)(-7,-15)(-11,18,-13)(-12,-18)(-14,7,-16,1,11)(2,16,6)(4,8,14,10)
Loop annotated with half-edges
11^1_28 annotated with half-edges