$12^{2}_{32}$ - Minimal pinning sets
Pinning sets for 12^2_32 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_32 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 240
of which optimal: 4
of which minimal: 4
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.97778
on average over minimal pinning sets: 2.3
on average over optimal pinning sets: 2.3
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {2, 4, 5, 8, 9}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {2, 4, 6, 8, 9}
5
[2, 2, 2, 2, 4]
2.40
C (optimal)
• {1, 2, 4, 8, 9}
5
[2, 2, 2, 2, 3]
2.20
D (optimal)
• {2, 3, 4, 8, 9}
5
[2, 2, 2, 2, 4]
2.40
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
4
0
0
2.3
6
0
0
22
2.62
7
0
0
52
2.84
8
0
0
69
3.0
9
0
0
56
3.11
10
0
0
28
3.2
11
0
0
8
3.27
12
0
0
1
3.33
Total
4
0
236
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,2,2],[0,1,1,6],[0,6,4,4],[0,3,3,5],[1,4,7,8],[2,8,7,3],[5,6,9,9],[5,9,9,6],[7,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[6,3,1,4],[4,7,5,20],[5,19,6,20],[12,2,13,3],[1,13,2,14],[7,14,8,15],[11,18,12,19],[8,18,9,17],[15,10,16,11],[9,16,10,17]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (7,6,-8,-1)(4,1,-5,-2)(10,19,-11,-20)(5,8,-6,-9)(16,9,-17,-10)(18,11,-19,-12)(12,17,-13,-18)(20,13,-7,-14)(3,14,-4,-15)(15,2,-16,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,4,14,-7)(-2,15,-4)(-3,-15)(-5,-9,16,2)(-6,7,13,17,9)(-8,5,1)(-10,-20,-14,3,-16)(-11,18,-13,20)(-12,-18)(-17,12,-19,10)(6,8)(11,19)
Multiloop annotated with half-edges
12^2_32 annotated with half-edges