$10^{1}_{27}$ - Minimal pinning sets
Pinning sets for 10^1_27 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 10^1_27 Pinning data
Pinning number of this loop: 4
Total number of pinning sets: 114
of which optimal: 1
of which minimal: 5
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.99877
on average over minimal pinning sets: 2.74
on average over optimal pinning sets: 2.5
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {2, 4, 5, 9}
4
[2, 2, 3, 3]
2.50
a (minimal)
• {2, 5, 6, 9, 10}
5
[2, 2, 3, 3, 4]
2.80
b (minimal)
• {2, 3, 5, 7, 9}
5
[2, 2, 3, 3, 4]
2.80
c (minimal)
• {2, 3, 5, 6, 9}
5
[2, 2, 3, 4, 4]
3.00
d (minimal)
• {1, 2, 5, 7, 10}
5
[2, 2, 3, 3, 3]
2.60
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
1
0
0
2.5
5
0
4
6
2.76
6
0
0
30
2.91
7
0
0
39
3.03
8
0
0
25
3.11
9
0
0
8
3.17
10
0
0
1
3.2
Total
1
4
109
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: [[1,2,2,3],[0,3,4,5],[0,5,6,0],[0,7,4,1],[1,3,7,5],[1,4,6,2],[2,5,7,7],[3,6,6,4]]
PD code (use to draw this loop with SnapPy): [[16,5,1,6],[6,12,7,11],[4,15,5,16],[1,13,2,12],[7,2,8,3],[3,10,4,11],[14,9,15,10],[13,9,14,8]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (5,16,-6,-1)(14,3,-15,-4)(6,11,-7,-12)(12,7,-13,-8)(1,8,-2,-9)(9,4,-10,-5)(10,15,-11,-16)(2,13,-3,-14)
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,-9,-5)(-2,-14,-4,9)(-3,14)(-6,-12,-8,1)(-7,12)(-10,-16,5)(-11,6,16)(-13,2,8)(-15,10,4)(3,13,7,11,15)
Loop annotated with half-edges
10^1_27 annotated with half-edges