$12^{2}_{255}$ - Minimal pinning sets
Pinning sets for 12^2_255 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_255 Pinning data
Pinning number of this multiloop: 6
Total number of pinning sets: 128
of which optimal: 3
of which minimal: 3
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.97249
on average over minimal pinning sets: 2.33333
on average over optimal pinning sets: 2.33333
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 6, 8, 9}
6
[2, 2, 2, 2, 3, 3]
2.33
B (optimal)
• {1, 2, 3, 6, 9, 10}
6
[2, 2, 2, 2, 3, 3]
2.33
C (optimal)
• {1, 2, 3, 7, 8, 9}
6
[2, 2, 2, 2, 3, 3]
2.33
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
6
3
0
0
2.33
7
0
0
16
2.64
8
0
0
35
2.87
9
0
0
40
3.04
10
0
0
25
3.18
11
0
0
8
3.27
12
0
0
1
3.33
Total
3
0
125
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 3, 3, 5, 5, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,2],[0,1,5,3],[0,2,6,4],[0,3,7,7],[1,8,2,1],[3,8,8,9],[4,9,9,4],[5,9,6,6],[6,8,7,7]]
PD code (use to draw this multiloop with SnapPy): [[14,20,1,15],[15,12,16,11],[13,10,14,11],[19,9,20,10],[1,9,2,8],[12,17,13,16],[18,4,19,5],[2,7,3,8],[17,6,18,5],[6,3,7,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (4,1,-5,-2)(15,2,-16,-3)(3,20,-4,-15)(14,5,-1,-6)(6,13,-7,-14)(16,7,-17,-8)(11,8,-12,-9)(9,18,-10,-19)(12,17,-13,-18)(19,10,-20,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,4,20,10,18,-13,6)(-2,15,-4)(-3,-15)(-5,14,-7,16,2)(-6,-14)(-8,11,-20,3,-16)(-9,-19,-11)(-10,19)(-12,-18,9)(-17,12,8)(1,5)(7,13,17)
Multiloop annotated with half-edges
12^2_255 annotated with half-edges