$12^{2}_{323}$ - Minimal pinning sets
Pinning sets for 12^2_323 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_323 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 288
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: 3.04277
on average over minimal pinning sets: 2.45
on average over optimal pinning sets: 2.45
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 5, 9, 10}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {1, 2, 5, 7, 10}
5
[2, 2, 2, 3, 4]
2.60
C (optimal)
• {1, 2, 5, 6, 10}
5
[2, 2, 2, 3, 3]
2.40
D (optimal)
• {1, 2, 5, 9, 12}
5
[2, 2, 2, 3, 3]
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
2.45
6
0
0
24
2.72
7
0
0
61
2.91
8
0
0
85
3.05
9
0
0
70
3.16
10
0
0
34
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
4
0
284
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],[0,4,5,2],[0,1,5,6],[0,6,6,7],[0,7,8,1],[1,9,9,2],[2,7,3,3],[3,6,8,4],[4,7,9,9],[5,8,8,5]]
PD code (use to draw this multiloop with SnapPy): [[6,20,1,7],[7,16,8,17],[17,5,18,6],[19,12,20,13],[1,15,2,16],[8,4,9,5],[18,14,19,13],[14,11,15,12],[2,11,3,10],[3,9,4,10]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (20,1,-7,-2)(15,2,-16,-3)(6,7,-1,-8)(8,5,-9,-6)(12,9,-13,-10)(17,10,-18,-11)(11,16,-12,-17)(4,13,-5,-14)(19,14,-20,-15)(3,18,-4,-19)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,20,14,-5,8)(-2,15,-20)(-3,-19,-15)(-4,-14,19)(-6,-8)(-7,6,-9,12,16,2)(-10,17,-12)(-11,-17)(-13,4,18,10)(-16,11,-18,3)(1,7)(5,13,9)
Multiloop annotated with half-edges
12^2_323 annotated with half-edges