$12^{1}_{132}$ - Minimal pinning sets
Pinning sets for 12^1_132 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_132 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, 5, 6, 8}
5
[2, 2, 2, 3, 3]
2.40
a (minimal)
• {1, 3, 4, 6, 7, 10}
6
[2, 2, 2, 3, 3, 3]
2.50
b (minimal)
• {1, 3, 5, 6, 7, 10}
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, 3, 4, 5, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,3,4,5],[0,6,3,0],[0,2,6,1],[1,7,8,5],[1,4,8,8],[2,9,7,3],[4,6,9,9],[4,9,5,5],[6,8,7,7]]
PD code (use to draw this loop with SnapPy): [[20,7,1,8],[8,18,9,17],[6,19,7,20],[1,19,2,18],[9,13,10,12],[16,11,17,12],[5,2,6,3],[13,5,14,4],[10,15,11,16],[3,15,4,14]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (1,10,-2,-11)(17,4,-18,-5)(13,8,-14,-9)(9,6,-10,-7)(11,20,-12,-1)(7,12,-8,-13)(14,5,-15,-6)(15,18,-16,-19)(3,16,-4,-17)(2,19,-3,-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,-11)(-2,-20,11)(-3,-17,-5,14,8,12,20)(-4,17)(-6,9,-14)(-7,-13,-9)(-8,13)(-10,1,-12,7)(-15,-19,2,10,6)(-16,3,19)(-18,15,5)(4,16,18)
Loop annotated with half-edges
12^1_132 annotated with half-edges