$12^{2}_{9}$ - Minimal pinning sets
Pinning sets for 12^2_9 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_9 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 128
of which optimal: 1
of which minimal: 1
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.90623
on average over minimal pinning sets: 2.0
on average over optimal pinning sets: 2.0
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 5, 7, 11}
5
[2, 2, 2, 2, 2]
2.00
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
1
0
0
2.0
6
0
0
7
2.38
7
0
0
21
2.65
8
0
0
35
2.86
9
0
0
35
3.02
10
0
0
21
3.14
11
0
0
7
3.25
12
0
0
1
3.33
Total
1
0
127
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 2, 3, 4, 4, 4, 4, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,4,4,5],[0,6,6,7],[0,8,4,0],[1,3,5,1],[1,4,8,9],[2,9,9,2],[2,9,8,8],[3,7,7,5],[5,7,6,6]]
PD code (use to draw this multiloop with SnapPy): [[16,20,1,17],[17,9,18,10],[15,4,16,5],[19,1,20,2],[8,18,9,19],[10,8,11,7],[5,14,6,15],[3,12,4,13],[2,12,3,11],[13,6,14,7]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (8,1,-9,-2)(2,5,-3,-6)(17,4,-18,-5)(6,13,-7,-14)(14,7,-15,-8)(16,9,-1,-10)(19,12,-20,-13)(10,15,-11,-16)(11,20,-12,-17)(3,18,-4,-19)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,8,-15,10)(-2,-6,-14,-8)(-3,-19,-13,6)(-4,17,-12,19)(-5,2,-9,16,-11,-17)(-7,14)(-10,-16)(-18,3,5)(-20,11,15,7,13)(1,9)(4,18)(12,20)
Multiloop annotated with half-edges
12^2_9 annotated with half-edges