$12^{1}_{213}$ - Minimal pinning sets
Pinning sets for 12^1_213 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_213 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 228
of which optimal: 2
of which minimal: 6
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.98552
on average over minimal pinning sets: 2.43333
on average over optimal pinning sets: 2.3
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 9, 11}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 2, 3, 7, 11}
5
[2, 2, 2, 2, 4]
2.40
a (minimal)
• {1, 2, 3, 4, 10, 11}
6
[2, 2, 2, 2, 3, 4]
2.50
b (minimal)
• {1, 2, 3, 5, 10, 11}
6
[2, 2, 2, 2, 4, 4]
2.67
c (minimal)
• {1, 2, 3, 4, 8, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
d (minimal)
• {1, 2, 3, 5, 8, 11}
6
[2, 2, 2, 2, 3, 4]
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
2.3
6
0
4
13
2.58
7
0
0
48
2.82
8
0
0
68
2.99
9
0
0
56
3.11
10
0
0
28
3.2
11
0
0
8
3.27
12
0
0
1
3.33
Total
2
4
222
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,2,0],[0,1,5,5],[0,5,6,7],[1,7,7,8],[2,9,3,2],[3,9,8,7],[3,6,4,4],[4,6,9,9],[5,8,8,6]]
PD code (use to draw this loop with SnapPy): [[13,20,14,1],[19,12,20,13],[14,12,15,11],[1,16,2,17],[7,18,8,19],[15,10,16,11],[2,5,3,6],[17,6,18,7],[8,3,9,4],[4,9,5,10]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (18,1,-19,-2)(13,2,-14,-3)(20,5,-1,-6)(11,6,-12,-7)(7,10,-8,-11)(15,8,-16,-9)(17,12,-18,-13)(3,14,-4,-15)(9,16,-10,-17)(4,19,-5,-20)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,18,12,6)(-2,13,-18)(-3,-15,-9,-17,-13)(-4,-20,-6,11,-8,15)(-5,20)(-7,-11)(-10,7,-12,17)(-14,3)(-16,9)(-19,4,14,2)(1,5,19)(8,10,16)
Loop annotated with half-edges
12^1_213 annotated with half-edges