$12^{2}_{139}$ - Minimal pinning sets
Pinning sets for 12^2_139 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_139 Pinning data
Pinning number of this multiloop: 6
Total number of pinning sets: 144
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: 2.97193
on average over minimal pinning sets: 2.33333
on average over optimal pinning sets: 2.33333
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 4, 7, 8}
6
[2, 2, 2, 2, 3, 3]
2.33
B (optimal)
• {1, 2, 3, 4, 7, 9}
6
[2, 2, 2, 2, 3, 3]
2.33
C (optimal)
• {2, 3, 4, 7, 8, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
D (optimal)
• {2, 3, 4, 7, 9, 11}
6
[2, 2, 2, 2, 3, 3]
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
2.33
7
0
0
20
2.66
8
0
0
41
2.89
9
0
0
44
3.06
10
0
0
26
3.18
11
0
0
8
3.27
12
0
0
1
3.33
Total
4
0
140
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 3, 5, 5, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,6],[0,7,8,8],[0,9,4,4],[0,3,3,1],[1,9,7,6],[1,5,7,7],[2,6,6,5],[2,9,9,2],[3,8,8,5]]
PD code (use to draw this multiloop with SnapPy): [[12,3,1,4],[4,13,5,20],[11,17,12,18],[2,14,3,15],[1,14,2,13],[5,9,6,8],[19,7,20,8],[18,7,19,6],[16,10,17,11],[15,10,16,9]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (10,1,-11,-2)(4,17,-5,-18)(18,5,-19,-6)(6,3,-7,-4)(7,14,-8,-15)(15,8,-16,-9)(9,16,-10,-17)(19,2,-20,-3)(20,11,-13,-12)(12,13,-1,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,10,16,8,14)(-2,19,5,17,-10)(-3,6,-19)(-4,-18,-6)(-5,18)(-7,-15,-9,-17,4)(-8,15)(-11,20,2)(-12,-14,7,3,-20)(-13,12)(-16,9)(1,13,11)
Multiloop annotated with half-edges
12^2_139 annotated with half-edges