$10^{1}_{7}$ - Minimal pinning sets
Pinning sets for 10^1_7 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 10^1_7 Pinning data
Pinning number of this loop: 4
Total number of pinning sets: 144
of which optimal: 4
of which minimal: 4
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.9857
on average over minimal pinning sets: 2.5625
on average over optimal pinning sets: 2.5625
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 5, 6}
4
[2, 2, 3, 3]
2.50
B (optimal)
• {1, 3, 5, 9}
4
[2, 2, 3, 3]
2.50
C (optimal)
• {1, 3, 5, 7}
4
[2, 2, 3, 4]
2.75
D (optimal)
• {1, 2, 5, 9}
4
[2, 2, 3, 3]
2.50
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
4
0
0
2.56
5
0
0
20
2.79
6
0
0
41
2.94
7
0
0
44
3.04
8
0
0
26
3.12
9
0
0
8
3.17
10
0
0
1
3.2
Total
4
0
140
Other information about this loop Properties
Region degree sequence: [2, 2, 3, 3, 3, 3, 3, 4, 4, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,3,2,0],[0,1,4,5],[0,5,6,1],[2,6,7,7],[2,7,6,3],[3,5,7,4],[4,6,5,4]]
PD code (use to draw this loop with SnapPy): [[9,16,10,1],[15,8,16,9],[10,8,11,7],[1,14,2,15],[11,5,12,4],[13,6,14,7],[2,6,3,5],[12,3,13,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (10,1,-11,-2)(2,9,-3,-10)(3,16,-4,-1)(11,4,-12,-5)(5,8,-6,-9)(13,6,-14,-7)(15,12,-16,-13)(7,14,-8,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,10,-3)(-2,-10)(-4,11,1)(-5,-9,2,-11)(-6,13,-16,3,9)(-7,-15,-13)(-8,5,-12,15)(-14,7)(4,16,12)(6,8,14)
Loop annotated with half-edges
10^1_7 annotated with half-edges