$12^{3}_{19}$ - Minimal pinning sets
Pinning sets for 12^3_19 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^3_19 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 379
of which optimal: 1
of which minimal: 7
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.0407
on average over minimal pinning sets: 2.61667
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)
• {2, 3, 7, 11}
4
[2, 2, 2, 3]
2.25
a (minimal)
• {3, 4, 5, 7, 11}
5
[2, 2, 2, 3, 4]
2.60
b (minimal)
• {1, 3, 5, 7, 11}
5
[2, 2, 2, 3, 3]
2.40
c (minimal)
• {3, 5, 7, 9, 11}
5
[2, 2, 2, 3, 5]
2.80
d (minimal)
• {3, 5, 6, 7, 11}
5
[2, 2, 2, 3, 4]
2.60
e (minimal)
• {3, 5, 7, 10, 11, 12}
6
[2, 2, 2, 3, 4, 5]
3.00
f (minimal)
• {3, 5, 7, 8, 10, 11}
6
[2, 2, 2, 3, 3, 4]
2.67
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
4
8
2.58
6
0
2
46
2.81
7
0
0
91
2.97
8
0
0
105
3.08
9
0
0
77
3.17
10
0
0
35
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
1
6
372
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,3,4,0],[0,4,4,5],[0,5,6,1],[1,7,2,2],[2,8,6,3],[3,5,9,7],[4,6,9,8],[5,7,9,9],[6,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[8,12,1,9],[9,7,10,8],[11,20,12,13],[1,6,2,7],[10,14,11,13],[5,19,6,20],[2,19,3,18],[14,18,15,17],[4,16,5,17],[3,16,4,15]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (7,4,-8,-5)(18,5,-19,-6)(6,17,-7,-18)(14,1,-15,-2)(2,15,-3,-16)(16,13,-9,-14)(9,8,-10,-1)(3,10,-4,-11)(20,11,-17,-12)(12,19,-13,-20)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,14,-9)(-2,-16,-14)(-3,-11,20,-13,16)(-4,7,17,11)(-5,18,-7)(-6,-18)(-8,9,13,19,5)(-10,3,15,1)(-12,-20)(-15,2)(-17,6,-19,12)(4,10,8)
Multiloop annotated with half-edges
12^3_19 annotated with half-edges