$12^{3}_{77}$ - Minimal pinning sets
Pinning sets for 12^3_77 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^3_77 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 336
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: 3.04109
on average over minimal pinning sets: 2.49444
on average over optimal pinning sets: 2.25
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 5}
4
[2, 2, 2, 3]
2.25
a (minimal)
• {1, 2, 3, 6, 9}
5
[2, 2, 2, 3, 3]
2.40
b (minimal)
• {1, 2, 3, 6, 7, 11}
6
[2, 2, 2, 3, 4, 4]
2.83
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
1
0
0
2.25
5
0
1
8
2.56
6
0
1
34
2.78
7
0
0
75
2.94
8
0
0
96
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
1
2
333
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 4, 5]
Minimal region degree: 2
Is multisimple: Yes
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,6,7],[0,7,8,3],[0,2,6,4],[0,3,5,5],[1,4,4,6],[1,5,3,8],[1,9,9,2],[2,9,9,6],[7,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[8,12,1,9],[9,3,10,4],[7,20,8,13],[11,19,12,20],[1,19,2,18],[2,17,3,18],[10,17,11,16],[4,14,5,13],[15,6,16,7],[14,6,15,5]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (18,1,-19,-2)(3,14,-4,-15)(15,4,-16,-5)(6,17,-7,-18)(16,19,-9,-20)(8,9,-1,-10)(10,7,-11,-8)(20,11,-17,-12)(5,12,-6,-13)(13,2,-14,-3)
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,-7,10)(-2,13,-6,-18)(-3,-15,-5,-13)(-4,15)(-8,-10)(-9,8,-11,20)(-12,5,-16,-20)(-14,3)(-17,6,12)(-19,16,4,14,2)(1,9,19)(7,17,11)
Multiloop annotated with half-edges
12^3_77 annotated with half-edges