$12^{2}_{300}$ - Minimal pinning sets
Pinning sets for 12^2_300 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_300 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 352
of which optimal: 5
of which minimal: 5
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.10848
on average over minimal pinning sets: 2.64
on average over optimal pinning sets: 2.64
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {2, 3, 4, 9, 11}
5
[2, 2, 3, 3, 3]
2.60
B (optimal)
• {2, 3, 4, 11, 12}
5
[2, 2, 3, 3, 3]
2.60
C (optimal)
• {1, 3, 4, 8, 12}
5
[2, 2, 3, 3, 3]
2.60
D (optimal)
• {1, 3, 4, 7, 12}
5
[2, 2, 3, 3, 4]
2.80
E (optimal)
• {1, 2, 3, 4, 12}
5
[2, 2, 3, 3, 3]
2.60
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
5
0
0
2.64
6
0
0
30
2.86
7
0
0
76
3.01
8
0
0
105
3.12
9
0
0
85
3.2
10
0
0
40
3.26
11
0
0
10
3.31
12
0
0
1
3.33
Total
5
0
347
Other information about this multiloop Properties
Region degree sequence: [2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 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],[0,6,7,7],[0,8,5,1],[1,4,8,2],[2,8,9,3],[3,9,9,3],[4,9,6,5],[6,8,7,7]]
PD code (use to draw this multiloop with SnapPy): [[4,20,1,5],[5,14,6,15],[15,3,16,4],[19,10,20,11],[1,13,2,14],[6,2,7,3],[16,12,17,11],[9,18,10,19],[12,7,13,8],[17,8,18,9]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (5,4,-6,-1)(12,3,-13,-4)(7,14,-8,-15)(9,16,-10,-17)(17,6,-18,-7)(2,19,-3,-20)(11,20,-12,-5)(15,8,-16,-9)(1,10,-2,-11)(18,13,-19,-14)
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,-11,-5)(-2,-20,11)(-3,12,20)(-4,5,-12)(-6,17,-10,1)(-7,-15,-9,-17)(-8,15)(-13,18,6,4)(-14,7,-18)(-16,9)(-19,2,10,16,8,14)(3,19,13)
Multiloop annotated with half-edges
12^2_300 annotated with half-edges