$12^{1}_{55}$ - Minimal pinning sets
Pinning sets for 12^1_55 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_55 Pinning data
Pinning number of this loop: 4
Total number of pinning sets: 505
of which optimal: 5
of which minimal: 9
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.04513
on average over minimal pinning sets: 2.51111
on average over optimal pinning sets: 2.6
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 4, 5, 7}
4
[2, 2, 2, 3]
2.25
B (optimal)
• {1, 5, 7, 9}
4
[2, 2, 2, 5]
2.75
C (optimal)
• {1, 5, 7, 11}
4
[2, 2, 2, 3]
2.25
D (optimal)
• {1, 5, 6, 7}
4
[2, 2, 2, 5]
2.75
E (optimal)
• {1, 5, 7, 12}
4
[2, 2, 2, 6]
3.00
a (minimal)
• {1, 5, 7, 8, 10}
5
[2, 2, 2, 3, 3]
2.40
b (minimal)
• {1, 3, 5, 7, 8}
5
[2, 2, 2, 3, 3]
2.40
c (minimal)
• {1, 2, 5, 7, 10}
5
[2, 2, 2, 3, 3]
2.40
d (minimal)
• {1, 2, 3, 5, 7}
5
[2, 2, 2, 3, 3]
2.40
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
5
0
0
2.6
5
0
4
30
2.73
6
0
0
84
2.89
7
0
0
126
3.02
8
0
0
126
3.11
9
0
0
84
3.19
10
0
0
36
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
5
4
496
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 3, 3, 5, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,4,5],[0,6,6,3],[0,2,7,8],[0,5,1,1],[1,4,9,6],[2,5,9,2],[3,9,8,8],[3,7,7,9],[5,8,7,6]]
PD code (use to draw this loop with SnapPy): [[3,20,4,1],[2,15,3,16],[19,8,20,9],[4,8,5,7],[1,17,2,16],[17,14,18,15],[9,18,10,19],[5,13,6,12],[6,11,7,12],[13,10,14,11]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (3,20,-4,-1)(15,4,-16,-5)(9,6,-10,-7)(2,7,-3,-8)(8,1,-9,-2)(13,10,-14,-11)(18,11,-19,-12)(12,17,-13,-18)(5,14,-6,-15)(19,16,-20,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,8,-3)(-2,-8)(-4,15,-6,9,1)(-5,-15)(-7,2,-9)(-10,13,17,-20,3,7)(-11,18,-13)(-12,-18)(-14,5,-16,19,11)(-17,12,-19)(4,20,16)(6,14,10)
Loop annotated with half-edges
12^1_55 annotated with half-edges