$12^{1}_{53}$ - Minimal pinning sets
Pinning sets for 12^1_53 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_53 Pinning data
Pinning number of this loop: 4
Total number of pinning sets: 396
of which optimal: 1
of which minimal: 6
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.05899
on average over minimal pinning sets: 2.50556
on average over optimal pinning sets: 2.5
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 8, 11}
4
[2, 2, 2, 4]
2.50
a (minimal)
• {1, 2, 5, 7, 11}
5
[2, 2, 2, 3, 3]
2.40
b (minimal)
• {1, 2, 5, 11, 12}
5
[2, 2, 2, 3, 3]
2.40
c (minimal)
• {1, 2, 7, 9, 11}
5
[2, 2, 2, 3, 3]
2.40
d (minimal)
• {1, 2, 3, 9, 11, 12}
6
[2, 2, 2, 3, 3, 3]
2.50
e (minimal)
• {1, 2, 6, 9, 11, 12}
6
[2, 2, 2, 3, 3, 5]
2.83
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
1
0
0
2.5
5
0
3
8
2.65
6
0
2
44
2.83
7
0
0
96
2.98
8
0
0
114
3.1
9
0
0
82
3.18
10
0
0
36
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
1
5
390
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: [[1,2,3,4],[0,4,4,5],[0,5,6,3],[0,2,7,7],[0,5,1,1],[1,4,6,2],[2,5,8,9],[3,9,8,3],[6,7,9,9],[6,8,8,7]]
PD code (use to draw this loop with SnapPy): [[3,20,4,1],[13,2,14,3],[19,10,20,11],[4,10,5,9],[1,12,2,13],[14,12,15,11],[15,18,16,19],[5,8,6,9],[6,17,7,18],[16,7,17,8]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (11,20,-12,-1)(14,5,-15,-6)(3,6,-4,-7)(7,2,-8,-3)(17,8,-18,-9)(9,12,-10,-13)(19,10,-20,-11)(4,15,-5,-16)(13,16,-14,-17)(1,18,-2,-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,-19,-11)(-2,7,-4,-16,13,-10,19)(-3,-7)(-5,14,16)(-6,3,-8,17,-14)(-9,-13,-17)(-12,9,-18,1)(-15,4,6)(-20,11)(2,18,8)(5,15)(10,12,20)
Loop annotated with half-edges
12^1_53 annotated with half-edges