$11^{1}_{69}$ - Minimal pinning sets
Pinning sets for 11^1_69 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^1_69 Pinning data
Pinning number of this loop: 6
Total number of pinning sets: 64
of which optimal: 3
of which minimal: 3
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.91137
on average over minimal pinning sets: 2.33333
on average over optimal pinning sets: 2.33333
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 5, 6, 10}
6
[2, 2, 2, 2, 3, 3]
2.33
B (optimal)
• {1, 2, 3, 5, 7, 10}
6
[2, 2, 2, 2, 3, 3]
2.33
C (optimal)
• {1, 2, 3, 4, 6, 10}
6
[2, 2, 2, 2, 3, 3]
2.33
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
6
3
0
0
2.33
7
0
0
13
2.66
8
0
0
22
2.9
9
0
0
18
3.07
10
0
0
7
3.2
11
0
0
1
3.27
Total
3
0
61
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 3, 4, 5, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,2,0],[0,1,5,3],[0,2,5,6],[1,7,7,5],[2,4,6,3],[3,5,8,8],[4,8,8,4],[6,7,7,6]]
PD code (use to draw this loop with SnapPy): [[18,9,1,10],[10,17,11,18],[11,8,12,9],[1,12,2,13],[16,3,17,4],[7,2,8,3],[13,7,14,6],[4,15,5,16],[14,5,15,6]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (10,1,-11,-2)(2,9,-3,-10)(7,4,-8,-5)(14,5,-15,-6)(15,8,-16,-9)(18,11,-1,-12)(6,13,-7,-14)(3,16,-4,-17)(12,17,-13,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,10,-3,-17,12)(-2,-10)(-4,7,13,17)(-5,14,-7)(-6,-14)(-8,15,5)(-9,2,-11,18,-13,6,-15)(-12,-18)(-16,3,9)(1,11)(4,16,8)
Loop annotated with half-edges
11^1_69 annotated with half-edges