$12^{2}_{73}$ - Minimal pinning sets
Pinning sets for 12^2_73 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_73 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 224
of which optimal: 1
of which minimal: 5
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.03933
on average over minimal pinning sets: 2.51333
on average over optimal pinning sets: 2.4
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {2, 3, 6, 9, 11}
5
[2, 2, 2, 3, 3]
2.40
a (minimal)
• {1, 2, 4, 6, 9, 11}
6
[2, 2, 2, 3, 3, 3]
2.50
b (minimal)
• {2, 3, 7, 8, 9, 11}
6
[2, 2, 2, 3, 3, 4]
2.67
c (minimal)
• {1, 2, 4, 8, 9, 11}
6
[2, 2, 2, 3, 3, 3]
2.50
d (minimal)
• {1, 2, 3, 8, 9, 11}
6
[2, 2, 2, 3, 3, 3]
2.50
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
1
0
0
2.4
6
0
4
7
2.62
7
0
0
39
2.84
8
0
0
67
3.01
9
0
0
63
3.14
10
0
0
33
3.23
11
0
0
9
3.29
12
0
0
1
3.33
Total
1
4
219
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: [[1,2,3,4],[0,4,2,2],[0,1,1,3],[0,2,5,6],[0,6,7,1],[3,7,8,8],[3,9,7,4],[4,6,9,5],[5,9,9,5],[6,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[10,7,1,8],[8,11,9,20],[9,19,10,20],[6,18,7,19],[1,12,2,11],[5,15,6,16],[17,12,18,13],[2,17,3,16],[14,4,15,5],[13,4,14,3]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (1,18,-2,-19)(19,2,-20,-3)(8,5,-9,-6)(3,20,-4,-11)(11,10,-12,-1)(4,13,-5,-14)(7,14,-8,-15)(15,6,-16,-7)(16,9,-17,-10)(12,17,-13,-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,-19,-3,-11)(-2,19)(-4,-14,7,-16,-10,11)(-5,8,14)(-6,15,-8)(-7,-15)(-9,16,6)(-12,-18,1)(-13,4,20,2,18)(-17,12,10)(-20,3)(5,13,17,9)
Multiloop annotated with half-edges
12^2_73 annotated with half-edges