$12^{3}_{8}$ - Minimal pinning sets
Pinning sets for 12^3_8 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^3_8 Pinning data
Pinning number of this multiloop: 6
Total number of pinning sets: 112
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: 2.91983
on average over minimal pinning sets: 2.22222
on average over optimal pinning sets: 2.22222
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 6, 8, 9}
6
[2, 2, 2, 2, 2, 3]
2.17
B (optimal)
• {1, 2, 3, 6, 9, 12}
6
[2, 2, 2, 2, 2, 4]
2.33
C (optimal)
• {1, 2, 3, 6, 9, 11}
6
[2, 2, 2, 2, 2, 3]
2.17
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
2.22
7
0
0
15
2.57
8
0
0
31
2.82
9
0
0
34
3.01
10
0
0
21
3.14
11
0
0
7
3.25
12
0
0
1
3.33
Total
3
0
109
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 2, 3, 3, 4, 4, 4, 6, 6]
Minimal region degree: 2
Is multisimple: Yes
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,5],[0,6,7,7],[0,7,7,6],[0,8,8,1],[1,8,9,1],[2,9,9,3],[2,3,3,2],[4,9,5,4],[5,8,6,6]]
PD code (use to draw this multiloop with SnapPy): [[8,16,1,9],[9,17,10,20],[13,7,14,8],[15,5,16,6],[1,18,2,17],[10,19,11,20],[4,12,5,13],[6,14,7,15],[18,3,19,2],[11,3,12,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (12,1,-13,-2)(10,3,-11,-4)(6,17,-7,-18)(16,7,-9,-8)(8,9,-1,-10)(2,11,-3,-12)(4,13,-5,-14)(15,18,-16,-19)(19,14,-20,-15)(20,5,-17,-6)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,12,-3,10)(-2,-12)(-4,-14,19,-16,-8,-10)(-5,20,14)(-6,-18,15,-20)(-7,16,18)(-9,8)(-11,2,-13,4)(-15,-19)(-17,6)(1,9,7,17,5,13)(3,11)
Multiloop annotated with half-edges
12^3_8 annotated with half-edges