$12^{2}_{170}$ - Minimal pinning sets
Pinning sets for 12^2_170 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_170 Pinning data
Pinning number of this multiloop: 5
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: 2.97631
on average over minimal pinning sets: 2.34444
on average over optimal pinning sets: 2.2
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 4, 7, 11}
5
[2, 2, 2, 2, 3]
2.20
a (minimal)
• {1, 2, 3, 4, 7, 9}
6
[2, 2, 2, 2, 3, 3]
2.33
b (minimal)
• {1, 3, 4, 7, 9, 12}
6
[2, 2, 2, 2, 3, 4]
2.50
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
1
0
0
2.2
6
0
2
7
2.5
7
0
0
30
2.75
8
0
0
51
2.94
9
0
0
49
3.08
10
0
0
27
3.19
11
0
0
8
3.27
12
0
0
1
3.33
Total
1
2
173
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 4, 4, 5, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,6],[0,7,7,3],[0,2,6,8],[0,8,9,9],[1,9,8,1],[1,3,7,7],[2,6,6,2],[3,5,9,4],[4,8,5,4]]
PD code (use to draw this multiloop with SnapPy): [[5,10,6,1],[4,20,5,11],[9,17,10,18],[6,17,7,16],[1,13,2,14],[11,3,12,4],[7,19,8,20],[18,8,19,9],[12,15,13,16],[2,15,3,14]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (18,1,-19,-2)(15,6,-16,-7)(3,8,-4,-9)(11,10,-12,-1)(19,12,-20,-13)(13,4,-14,-5)(7,14,-8,-15)(5,16,-6,-17)(2,17,-3,-18)(9,20,-10,-11)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,18,-3,-9,-11)(-2,-18)(-4,13,-20,9)(-5,-17,2,-19,-13)(-6,15,-8,3,17)(-7,-15)(-10,11)(-12,19,1)(-14,7,-16,5)(4,8,14)(6,16)(10,20,12)
Multiloop annotated with half-edges
12^2_170 annotated with half-edges