$10^{1}_{15}$ - Minimal pinning sets
Pinning sets for 10^1_15 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 10^1_15 Pinning data
Pinning number of this loop: 4
Total number of pinning sets: 112
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: 2.90574
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}
4
[2, 2, 2, 3]
2.25
B (optimal)
• {1, 3, 5, 10}
4
[2, 2, 2, 3]
2.25
C (optimal)
• {1, 3, 5, 9}
4
[2, 2, 2, 4]
2.50
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
2.33
5
0
0
15
2.64
6
0
0
31
2.84
7
0
0
34
2.97
8
0
0
21
3.07
9
0
0
7
3.14
10
0
0
1
3.2
Total
3
0
109
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 3, 3, 4, 4, 4, 4, 4]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,4,4,5],[0,5,6,0],[0,6,7,4],[1,3,5,1],[1,4,7,2],[2,7,7,3],[3,6,6,5]]
PD code (use to draw this loop with SnapPy): [[7,16,8,1],[11,6,12,7],[15,8,16,9],[1,4,2,5],[5,10,6,11],[12,10,13,9],[3,14,4,15],[2,14,3,13]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (15,2,-16,-3)(9,4,-10,-5)(13,6,-14,-7)(7,10,-8,-11)(3,8,-4,-9)(11,16,-12,-1)(1,12,-2,-13)(5,14,-6,-15)
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)
Face permutation $\varphi=(\sigma\epsilon)^{-1}=$ (-1,-13,-7,-11)(-2,15,-6,13)(-3,-9,-5,-15)(-4,9)(-8,3,-16,11)(-10,7,-14,5)(-12,1)(2,12,16)(4,8,10)(6,14)
Loop annotated with half-edges
10^1_15 annotated with half-edges