$12^{2}_{202}$ - Minimal pinning sets
Pinning sets for 12^2_202 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_202 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 272
of which optimal: 3
of which minimal: 4
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.0386
on average over minimal pinning sets: 2.46667
on average over optimal pinning sets: 2.4
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 5, 7, 11}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {1, 3, 4, 7, 11}
5
[2, 2, 2, 3, 3]
2.40
C (optimal)
• {1, 2, 3, 5, 7}
5
[2, 2, 2, 3, 3]
2.40
a (minimal)
• {1, 2, 3, 4, 7, 12}
6
[2, 2, 2, 3, 3, 4]
2.67
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
2.4
6
0
1
19
2.68
7
0
0
55
2.89
8
0
0
81
3.04
9
0
0
69
3.15
10
0
0
34
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
3
1
268
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,2,0],[0,1,5,3],[0,2,6,4],[1,3,7,7],[2,8,9,6],[3,5,9,7],[4,6,8,4],[5,7,9,9],[5,8,8,6]]
PD code (use to draw this multiloop with SnapPy): [[8,20,1,9],[9,7,10,8],[10,19,11,20],[1,11,2,12],[12,6,13,7],[18,15,19,16],[2,15,3,14],[5,13,6,14],[16,5,17,4],[17,3,18,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (8,9,-1,-10)(17,2,-18,-3)(14,3,-15,-4)(10,7,-11,-8)(5,16,-6,-17)(1,18,-2,-19)(19,6,-20,-7)(11,20,-12,-9)(15,12,-16,-13)(4,13,-5,-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,-19,-7,10)(-2,17,-6,19)(-3,14,-5,-17)(-4,-14)(-8,-10)(-9,8,-11)(-12,15,3,-18,1,9)(-13,4,-15)(-16,5,13)(-20,11,7)(2,18)(6,16,12,20)
Multiloop annotated with half-edges
12^2_202 annotated with half-edges