$12^{3}_{79}$ - Minimal pinning sets
Pinning sets for 12^3_79 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^3_79 Pinning data
Pinning number of this multiloop: 6
Total number of pinning sets: 128
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: 2.97249
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, 6, 8, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
B (optimal)
• {1, 2, 3, 5, 8, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
C (optimal)
• {1, 2, 3, 5, 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
3
0
0
2.33
7
0
0
16
2.64
8
0
0
35
2.87
9
0
0
40
3.04
10
0
0
25
3.18
11
0
0
8
3.27
12
0
0
1
3.33
Total
3
0
125
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: Yes
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,6],[0,7,7,3],[0,2,8,4],[0,3,5,1],[1,4,9,9],[1,8,7,7],[2,6,6,2],[3,6,9,9],[5,8,8,5]]
PD code (use to draw this multiloop with SnapPy): [[8,14,1,9],[9,15,10,20],[7,17,8,18],[13,16,14,17],[1,16,2,15],[10,2,11,3],[5,19,6,20],[18,6,19,7],[12,4,13,5],[11,4,12,3]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (8,9,-1,-10)(10,1,-11,-2)(20,3,-15,-4)(14,7,-9,-8)(5,12,-6,-13)(11,18,-12,-19)(4,19,-5,-20)(2,15,-3,-16)(16,13,-17,-14)(17,6,-18,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,10)(-2,-16,-14,-8,-10)(-3,20,-5,-13,16)(-4,-20)(-6,17,13)(-7,14,-17)(-9,8)(-11,-19,4,-15,2)(-12,5,19)(-18,11,1,9,7)(3,15)(6,12,18)
Multiloop annotated with half-edges
12^3_79 annotated with half-edges