$12^{1}_{19}$ - Minimal pinning sets
Pinning sets for 12^1_19 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_19 Pinning data
Pinning number of this loop: 4
Total number of pinning sets: 608
of which optimal: 5
of which minimal: 5
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.11671
on average over minimal pinning sets: 2.6
on average over optimal pinning sets: 2.6
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 4, 7, 10}
4
[2, 2, 3, 3]
2.50
B (optimal)
• {1, 4, 7, 9}
4
[2, 2, 3, 4]
2.75
C (optimal)
• {1, 4, 5, 7}
4
[2, 2, 3, 4]
2.75
D (optimal)
• {1, 2, 3, 7}
4
[2, 2, 3, 3]
2.50
E (optimal)
• {1, 3, 4, 7}
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
5
0
0
2.6
5
0
0
33
2.83
6
0
0
95
2.98
7
0
0
156
3.09
8
0
0
160
3.17
9
0
0
105
3.23
10
0
0
43
3.27
11
0
0
10
3.31
12
0
0
1
3.33
Total
5
0
603
Other information about this loop Properties
Region degree sequence: [2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,3,4,5],[0,5,4,3],[0,2,1,0],[1,2,6,7],[1,7,8,2],[4,8,9,9],[4,9,8,5],[5,7,9,6],[6,8,7,6]]
PD code (use to draw this loop with SnapPy): [[9,20,10,1],[11,8,12,9],[19,2,20,3],[10,2,11,1],[7,18,8,19],[12,4,13,3],[6,15,7,16],[17,4,18,5],[13,17,14,16],[14,5,15,6]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (20,9,-1,-10)(14,1,-15,-2)(11,2,-12,-3)(8,5,-9,-6)(3,12,-4,-13)(13,10,-14,-11)(4,15,-5,-16)(6,17,-7,-18)(18,7,-19,-8)(16,19,-17,-20)
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)(-19,19)(-20,20)
Face permutation $\varphi=(\sigma\epsilon)^{-1}=$ (-1,14,10)(-2,11,-14)(-3,-13,-11)(-4,-16,-20,-10,13)(-5,8,-19,16)(-6,-18,-8)(-7,18)(-9,20,-17,6)(-12,3)(-15,4,12,2)(1,9,5,15)(7,17,19)
Loop annotated with half-edges
12^1_19 annotated with half-edges