$12^{2}_{239}$ - Minimal pinning sets
Pinning sets for 12^2_239 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_239 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 192
of which optimal: 2
of which minimal: 2
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.96906
on average over minimal pinning sets: 2.2
on average over optimal pinning sets: 2.2
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, 2, 3]
2.20
B (optimal)
• {1, 2, 5, 7, 11}
5
[2, 2, 2, 2, 3]
2.20
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
2
0
0
2.2
6
0
0
13
2.54
7
0
0
36
2.78
8
0
0
55
2.95
9
0
0
50
3.09
10
0
0
27
3.19
11
0
0
8
3.27
12
0
0
1
3.33
Total
2
0
190
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,2,2],[0,1,1,3],[0,2,6,7],[0,7,7,8],[1,8,8,6],[3,5,9,9],[3,9,4,4],[4,9,5,5],[6,8,7,6]]
PD code (use to draw this multiloop with SnapPy): [[16,13,1,14],[14,6,15,5],[15,4,16,5],[12,3,13,4],[1,9,2,8],[6,17,7,20],[11,19,12,20],[2,9,3,10],[7,17,8,18],[18,10,19,11]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (10,1,-11,-2)(13,2,-14,-3)(3,12,-4,-13)(6,15,-7,-16)(16,7,-1,-8)(19,8,-20,-9)(14,11,-15,-12)(17,4,-18,-5)(5,18,-6,-19)(9,20,-10,-17)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,10,20,8)(-2,13,-4,17,-10)(-3,-13)(-5,-19,-9,-17)(-6,-16,-8,19)(-7,16)(-11,14,2)(-12,3,-14)(-15,6,18,4,12)(-18,5)(-20,9)(1,7,15,11)
Multiloop annotated with half-edges
12^2_239 annotated with half-edges