$11^{1}_{48}$ - Minimal pinning sets
Pinning sets for 11^1_48 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^1_48 Pinning data
Pinning number of this loop: 4
Total number of pinning sets: 176
of which optimal: 1
of which minimal: 3
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.97775
on average over minimal pinning sets: 2.41667
on average over optimal pinning sets: 2.25
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 10}
4
[2, 2, 2, 3]
2.25
a (minimal)
• {1, 2, 4, 7, 10}
5
[2, 2, 2, 3, 4]
2.60
b (minimal)
• {1, 2, 4, 6, 10}
5
[2, 2, 2, 3, 3]
2.40
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
2.25
5
0
2
7
2.56
6
0
0
30
2.78
7
0
0
51
2.95
8
0
0
49
3.07
9
0
0
27
3.16
10
0
0
8
3.23
11
0
0
1
3.27
Total
1
2
173
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,2],[0,1,5,6],[0,6,7,7],[0,8,6,5],[1,4,2,1],[2,4,8,3],[3,8,8,3],[4,7,7,6]]
PD code (use to draw this loop with SnapPy): [[9,18,10,1],[8,15,9,16],[17,14,18,15],[10,3,11,4],[1,6,2,7],[16,7,17,8],[2,13,3,14],[11,5,12,4],[12,5,13,6]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (18,7,-1,-8)(8,1,-9,-2)(15,2,-16,-3)(12,3,-13,-4)(16,9,-17,-10)(13,10,-14,-11)(4,11,-5,-12)(5,14,-6,-15)(6,17,-7,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,8)(-2,15,-6,-18,-8)(-3,12,-5,-15)(-4,-12)(-7,18)(-9,16,2)(-10,13,3,-16)(-11,4,-13)(-14,5,11)(-17,6,14,10)(1,7,17,9)
Loop annotated with half-edges
11^1_48 annotated with half-edges