$12^{2}_{77}$ - Minimal pinning sets
Pinning sets for 12^2_77 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_77 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 320
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: 3.04931
on average over minimal pinning sets: 2.48
on average over optimal pinning sets: 2.48
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 5, 6, 7}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {1, 2, 5, 7, 10}
5
[2, 2, 2, 3, 3]
2.40
C (optimal)
• {1, 2, 3, 5, 7}
5
[2, 2, 2, 3, 3]
2.40
D (optimal)
• {1, 2, 5, 7, 12}
5
[2, 2, 2, 3, 4]
2.60
E (optimal)
• {1, 3, 4, 5, 7}
5
[2, 2, 2, 3, 4]
2.60
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
5
0
0
2.48
6
0
0
29
2.75
7
0
0
71
2.93
8
0
0
95
3.07
9
0
0
75
3.17
10
0
0
35
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
5
0
315
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,3,2,0],[0,1,4,5],[0,5,6,1],[2,7,7,8],[2,9,6,3],[3,5,9,7],[4,6,8,4],[4,7,9,9],[5,8,8,6]]
PD code (use to draw this multiloop with SnapPy): [[12,20,1,13],[13,11,14,12],[14,19,15,20],[1,10,2,11],[6,18,7,19],[15,9,16,10],[2,16,3,17],[17,5,18,6],[7,5,8,4],[8,3,9,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (6,1,-7,-2)(2,5,-3,-6)(14,3,-15,-4)(20,7,-13,-8)(11,8,-12,-9)(18,9,-19,-10)(12,13,-1,-14)(4,15,-5,-16)(19,16,-20,-17)(10,17,-11,-18)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,6,-3,14)(-2,-6)(-4,-16,19,9,-12,-14)(-5,2,-7,20,16)(-8,11,17,-20)(-9,18,-11)(-10,-18)(-13,12,8)(-15,4)(-17,10,-19)(1,13,7)(3,5,15)
Multiloop annotated with half-edges
12^2_77 annotated with half-edges