$12^{3}_{10}$ - Minimal pinning sets
Pinning sets for 12^3_10 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^3_10 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 224
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: 2.9785
on average over minimal pinning sets: 2.26667
on average over optimal pinning sets: 2.26667
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 4, 6, 8, 9}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 4, 6, 9, 12}
5
[2, 2, 2, 2, 4]
2.40
C (optimal)
• {1, 4, 6, 9, 11}
5
[2, 2, 2, 2, 3]
2.20
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
3
0
0
2.27
6
0
0
18
2.59
7
0
0
46
2.82
8
0
0
65
2.98
9
0
0
55
3.11
10
0
0
28
3.2
11
0
0
8
3.27
12
0
0
1
3.33
Total
3
0
221
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,5],[0,6,6,7],[0,7,7,8],[0,9,9,1],[1,9,8,1],[2,8,7,2],[2,6,3,3],[3,6,5,9],[4,8,5,4]]
PD code (use to draw this multiloop with SnapPy): [[6,16,1,7],[7,17,8,20],[13,5,14,6],[15,10,16,11],[1,18,2,17],[8,19,9,20],[4,12,5,13],[14,12,15,11],[9,3,10,4],[18,3,19,2]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (8,1,-9,-2)(4,17,-5,-18)(14,5,-15,-6)(10,15,-11,-16)(16,9,-7,-10)(6,7,-1,-8)(2,11,-3,-12)(13,18,-14,-19)(19,12,-20,-13)(20,3,-17,-4)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,8)(-2,-12,19,-14,-6,-8)(-3,20,12)(-4,-18,13,-20)(-5,14,18)(-7,6,-15,10)(-9,16,-11,2)(-10,-16)(-13,-19)(-17,4)(1,7,9)(3,11,15,5,17)
Multiloop annotated with half-edges
12^3_10 annotated with half-edges