$10^{1}_{9}$ - Minimal pinning sets
Pinning sets for 10^1_9 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 10^1_9 Pinning data
Pinning number of this loop: 6
Total number of pinning sets: 20
of which optimal: 1
of which minimal: 2
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.78143
on average over minimal pinning sets: 2.22619
on average over optimal pinning sets: 2.16667
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 4, 7, 8}
6
[2, 2, 2, 2, 2, 3]
2.17
a (minimal)
• {1, 2, 3, 4, 6, 7, 9}
7
[2, 2, 2, 2, 2, 3, 3]
2.29
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
6
1
0
0
2.17
7
0
1
4
2.49
8
0
0
8
2.81
9
0
0
5
3.07
10
0
0
1
3.2
Total
1
1
18
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 2, 3, 3, 3, 6, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,2],[0,3,3,0],[0,4,4,0],[1,5,6,1],[2,6,7,2],[3,7,7,6],[3,5,7,4],[4,6,5,5]]
PD code (use to draw this loop with SnapPy): [[7,16,8,1],[15,6,16,7],[8,2,9,1],[5,14,6,15],[2,10,3,9],[4,11,5,12],[13,10,14,11],[3,13,4,12]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (10,1,-11,-2)(4,13,-5,-14)(8,5,-9,-6)(15,6,-16,-7)(7,14,-8,-15)(16,9,-1,-10)(2,11,-3,-12)(12,3,-13,-4)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,10)(-2,-12,-4,-14,7,-16,-10)(-3,12)(-5,8,14)(-6,15,-8)(-7,-15)(-9,16,6)(-11,2)(-13,4)(1,9,5,13,3,11)
Loop annotated with half-edges
10^1_9 annotated with half-edges