$12^{1}_{321}$ - Minimal pinning sets
Pinning sets for 12^1_321 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_321 Pinning data
Pinning number of this loop: 4
Total number of pinning sets: 504
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: 3.11974
on average over minimal pinning sets: 2.675
on average over optimal pinning sets: 2.625
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {2, 3, 7, 11}
4
[2, 2, 3, 3]
2.50
B (optimal)
• {2, 3, 10, 11}
4
[2, 2, 3, 4]
2.75
a (minimal)
• {2, 3, 9, 11, 12}
5
[2, 2, 3, 3, 3]
2.60
b (minimal)
• {2, 3, 4, 11, 12}
5
[2, 2, 3, 3, 3]
2.60
c (minimal)
• {2, 3, 8, 11, 12}
5
[2, 2, 3, 3, 5]
3.00
d (minimal)
• {1, 2, 3, 9, 12}
5
[2, 2, 3, 3, 3]
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
2.62
5
0
4
15
2.81
6
0
0
67
2.96
7
0
0
125
3.07
8
0
0
140
3.16
9
0
0
98
3.22
10
0
0
42
3.27
11
0
0
10
3.31
12
0
0
1
3.33
Total
2
4
498
Other information about this loop Properties
Region degree sequence: [2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,6],[0,7,8,3],[0,2,9,9],[0,9,5,1],[1,4,9,6],[1,5,8,7],[2,6,8,8],[2,7,7,6],[3,5,4,3]]
PD code (use to draw this loop with SnapPy): [[20,11,1,12],[12,8,13,7],[19,16,20,17],[10,15,11,16],[1,9,2,8],[13,2,14,3],[3,6,4,7],[17,4,18,5],[5,18,6,19],[14,9,15,10]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (11,20,-12,-1)(15,2,-16,-3)(1,4,-2,-5)(5,10,-6,-11)(6,19,-7,-20)(12,7,-13,-8)(18,9,-19,-10)(8,13,-9,-14)(17,14,-18,-15)(3,16,-4,-17)
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,-5,-11)(-2,15,-18,-10,5)(-3,-17,-15)(-4,1,-12,-8,-14,17)(-6,-20,11)(-7,12,20)(-9,18,14)(-13,8)(-16,3)(-19,6,10)(2,4,16)(7,19,9,13)
Loop annotated with half-edges
12^1_321 annotated with half-edges