$12^{2}_{258}$ - Minimal pinning sets
Pinning sets for 12^2_258 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_258 Pinning data
Pinning number of this multiloop: 6
Total number of pinning sets: 112
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: 2.91983
on average over minimal pinning sets: 2.22222
on average over optimal pinning sets: 2.22222
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 5, 6, 11}
6
[2, 2, 2, 2, 2, 3]
2.17
B (optimal)
• {1, 2, 3, 5, 9, 11}
6
[2, 2, 2, 2, 2, 3]
2.17
C (optimal)
• {1, 2, 3, 5, 7, 11}
6
[2, 2, 2, 2, 2, 4]
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
2.22
7
0
0
15
2.57
8
0
0
31
2.82
9
0
0
34
3.01
10
0
0
21
3.14
11
0
0
7
3.25
12
0
0
1
3.33
Total
3
0
109
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 2, 3, 3, 4, 4, 4, 6, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,4,4,5],[0,6,6,7],[0,7,7,0],[1,8,5,1],[1,4,9,9],[2,9,8,2],[2,8,3,3],[4,7,6,9],[5,8,6,5]]
PD code (use to draw this multiloop with SnapPy): [[12,20,1,13],[13,7,14,8],[17,11,18,12],[19,1,20,2],[6,14,7,15],[8,6,9,5],[10,16,11,17],[18,3,19,2],[15,3,16,4],[9,4,10,5]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (12,13,-1,-14)(18,1,-19,-2)(7,2,-8,-3)(3,6,-4,-7)(16,5,-17,-6)(20,9,-13,-10)(14,11,-15,-12)(4,17,-5,-18)(8,19,-9,-20)(10,15,-11,-16)
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,-5,16,-11,14)(-2,7,-4,-18)(-3,-7)(-6,3,-8,-20,-10,-16)(-9,20)(-12,-14)(-13,12,-15,10)(-17,4,6)(-19,8,2)(1,13,9,19)(5,17)(11,15)
Multiloop annotated with half-edges
12^2_258 annotated with half-edges