$12^{1}_{83}$ - Minimal pinning sets
Pinning sets for 12^1_83 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_83 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 240
of which optimal: 2
of which minimal: 4
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.03455
on average over minimal pinning sets: 2.45
on average over optimal pinning sets: 2.4
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 4, 6, 8}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {1, 3, 4, 6, 10}
5
[2, 2, 2, 3, 3]
2.40
a (minimal)
• {1, 2, 4, 6, 8, 11}
6
[2, 2, 2, 3, 3, 3]
2.50
b (minimal)
• {1, 2, 4, 6, 10, 11}
6
[2, 2, 2, 3, 3, 3]
2.50
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
2.4
6
0
2
13
2.66
7
0
0
45
2.86
8
0
0
71
3.02
9
0
0
64
3.14
10
0
0
33
3.23
11
0
0
9
3.29
12
0
0
1
3.33
Total
2
2
236
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,2,2],[0,1,1,5],[0,5,5,6],[0,6,7,1],[2,8,3,3],[3,9,7,4],[4,6,9,8],[5,7,9,9],[6,8,8,7]]
PD code (use to draw this loop with SnapPy): [[20,17,1,18],[18,13,19,14],[14,19,15,20],[16,7,17,8],[1,12,2,13],[15,9,16,8],[11,6,12,7],[2,6,3,5],[9,5,10,4],[10,3,11,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (8,1,-9,-2)(2,7,-3,-8)(3,20,-4,-1)(9,4,-10,-5)(14,5,-15,-6)(6,13,-7,-14)(19,10,-20,-11)(16,11,-17,-12)(18,15,-19,-16)(12,17,-13,-18)
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,-3)(-2,-8)(-4,9,1)(-5,14,-7,2,-9)(-6,-14)(-10,19,15,5)(-11,16,-19)(-12,-18,-16)(-13,6,-15,18)(-17,12)(-20,3,7,13,17,11)(4,20,10)
Loop annotated with half-edges
12^1_83 annotated with half-edges