$12^{1}_{340}$ - Minimal pinning sets
Pinning sets for 12^1_340 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_340 Pinning data
Pinning number of this loop: 4
Total number of pinning sets: 432
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: 3.05229
on average over minimal pinning sets: 2.4375
on average over optimal pinning sets: 2.375
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 5, 7, 11}
4
[2, 2, 2, 3]
2.25
B (optimal)
• {1, 5, 8, 11}
4
[2, 2, 2, 4]
2.50
a (minimal)
• {1, 4, 5, 10, 11}
5
[2, 2, 2, 3, 3]
2.40
b (minimal)
• {1, 4, 5, 9, 11}
5
[2, 2, 2, 3, 4]
2.60
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
2
0
0
2.38
5
0
2
15
2.65
6
0
0
58
2.85
7
0
0
107
2.99
8
0
0
119
3.1
9
0
0
83
3.18
10
0
0
36
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
2
2
428
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,6],[0,6,6,3],[0,2,7,8],[0,8,8,1],[1,9,7,6],[1,5,2,2],[3,5,9,9],[3,9,4,4],[5,8,7,7]]
PD code (use to draw this loop with SnapPy): [[11,20,12,1],[17,10,18,11],[19,4,20,5],[12,4,13,3],[1,16,2,17],[9,6,10,7],[18,6,19,5],[13,9,14,8],[15,2,16,3],[7,15,8,14]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (13,20,-14,-1)(1,10,-2,-11)(17,2,-18,-3)(3,16,-4,-17)(7,4,-8,-5)(6,11,-7,-12)(15,8,-16,-9)(12,5,-13,-6)(19,14,-20,-15)(9,18,-10,-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,-11,6,-13)(-2,17,-4,7,11)(-3,-17)(-5,12,-7)(-6,-12)(-8,15,-20,13,5)(-9,-19,-15)(-10,1,-14,19)(-16,3,-18,9)(2,10,18)(4,16,8)(14,20)
Loop annotated with half-edges
12^1_340 annotated with half-edges