$12^{2}_{44}$ - Minimal pinning sets
Pinning sets for 12^2_44 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_44 Pinning data
Pinning number of this multiloop: 6
Total number of pinning sets: 120
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: 2.91853
on average over minimal pinning sets: 2.25
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)
• {1, 2, 3, 4, 9, 11}
6
[2, 2, 2, 2, 2, 3]
2.17
B (optimal)
• {1, 2, 3, 4, 8, 11}
6
[2, 2, 2, 2, 2, 4]
2.33
C (optimal)
• {1, 2, 3, 4, 7, 11}
6
[2, 2, 2, 2, 2, 3]
2.17
D (optimal)
• {1, 2, 3, 4, 10, 11}
6
[2, 2, 2, 2, 2, 4]
2.33
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
6
4
0
0
2.25
7
0
0
18
2.6
8
0
0
34
2.85
9
0
0
35
3.02
10
0
0
21
3.14
11
0
0
7
3.25
12
0
0
1
3.33
Total
4
0
116
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 2, 3, 3, 4, 4, 5, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,6],[0,6,7,7],[0,8,8,6],[0,9,9,1],[1,9,9,6],[1,5,3,2],[2,8,8,2],[3,7,7,3],[4,5,5,4]]
PD code (use to draw this multiloop with SnapPy): [[5,10,6,1],[4,20,5,11],[15,9,16,10],[6,18,7,19],[1,12,2,11],[13,3,14,4],[14,19,15,20],[8,16,9,17],[17,7,18,8],[12,3,13,2]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (11,10,-12,-1)(15,4,-16,-5)(13,6,-14,-7)(2,7,-3,-8)(9,20,-10,-11)(5,14,-6,-15)(3,16,-4,-17)(12,17,-13,-18)(1,18,-2,-19)(19,8,-20,-9)
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,-9,-11)(-2,-8,19)(-3,-17,12,10,20,8)(-4,15,-6,13,17)(-5,-15)(-7,2,18,-13)(-10,11)(-12,-18,1)(-14,5,-16,3,7)(-20,9)(4,16)(6,14)
Multiloop annotated with half-edges
12^2_44 annotated with half-edges