$12^{2}_{206}$ - Minimal pinning sets
Pinning sets for 12^2_206 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_206 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 256
of which optimal: 3
of which minimal: 3
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.0346
on average over minimal pinning sets: 2.4
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, 4, 5, 9, 11}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {1, 4, 5, 10, 11}
5
[2, 2, 2, 3, 3]
2.40
C (optimal)
• {1, 2, 4, 5, 10}
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
3
0
0
2.4
6
0
0
19
2.68
7
0
0
51
2.89
8
0
0
75
3.03
9
0
0
65
3.15
10
0
0
33
3.23
11
0
0
9
3.29
12
0
0
1
3.33
Total
3
0
253
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 3, 4, 5, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,2],[0,1,6,7],[0,8,9,4],[0,3,9,6],[1,6,6,1],[2,5,5,4],[2,9,8,8],[3,7,7,9],[3,8,7,4]]
PD code (use to draw this multiloop with SnapPy): [[12,7,1,8],[8,13,9,20],[11,19,12,20],[6,16,7,17],[1,16,2,15],[13,10,14,9],[14,10,15,11],[18,4,19,5],[17,4,18,3],[5,2,6,3]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (3,12,-4,-1)(1,20,-2,-13)(13,2,-14,-3)(10,5,-11,-6)(7,16,-8,-17)(17,8,-18,-9)(18,11,-19,-12)(4,19,-5,-20)(9,14,-10,-15)(15,6,-16,-7)
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,-13,-3)(-2,13)(-4,-20,1)(-5,10,14,2,20)(-6,15,-10)(-7,-17,-9,-15)(-8,17)(-11,18,8,16,6)(-12,3,-14,9,-18)(-16,7)(-19,4,12)(5,19,11)
Multiloop annotated with half-edges
12^2_206 annotated with half-edges