$12^{2}_{256}$ - Minimal pinning sets
Pinning sets for 12^2_256 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_256 Pinning data
Pinning number of this multiloop: 6
Total number of pinning sets: 136
of which optimal: 3
of which minimal: 4
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.97614
on average over minimal pinning sets: 2.39286
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, 3, 5, 6, 7, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
B (optimal)
• {1, 3, 4, 6, 7, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
C (optimal)
• {3, 4, 6, 7, 11, 12}
6
[2, 2, 2, 2, 3, 3]
2.33
a (minimal)
• {2, 3, 5, 6, 7, 11, 12}
7
[2, 2, 2, 2, 3, 3, 4]
2.57
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
1
16
2.64
8
0
0
38
2.88
9
0
0
43
3.05
10
0
0
26
3.18
11
0
0
8
3.27
12
0
0
1
3.33
Total
3
1
132
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 6, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,4,0],[0,4,5,3],[0,2,5,6],[1,7,2,1],[2,8,8,3],[3,8,9,7],[4,6,9,9],[5,9,6,5],[6,8,7,7]]
PD code (use to draw this multiloop with SnapPy): [[12,20,1,13],[13,11,14,12],[19,9,20,10],[1,9,2,8],[10,14,11,15],[18,2,19,3],[4,7,5,8],[15,5,16,6],[3,17,4,18],[6,16,7,17]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (19,2,-20,-3)(1,4,-2,-5)(5,10,-6,-11)(15,6,-16,-7)(17,8,-18,-9)(14,11,-15,-12)(3,20,-4,-13)(12,13,-1,-14)(9,16,-10,-17)(7,18,-8,-19)
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,-5,-11,14)(-2,19,-8,17,-10,5)(-3,-13,12,-15,-7,-19)(-4,1,13)(-6,15,11)(-9,-17)(-12,-14)(-16,9,-18,7)(-20,3)(2,4,20)(6,10,16)(8,18)
Multiloop annotated with half-edges
12^2_256 annotated with half-edges