$12^{1}_{204}$ - Minimal pinning sets
Pinning sets for 12^1_204 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_204 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 296
of which optimal: 1
of which minimal: 9
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.10506
on average over minimal pinning sets: 2.6963
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, 3, 4, 6, 9}
5
[2, 2, 3, 3, 3]
2.60
a (minimal)
• {1, 3, 4, 5, 7, 9}
6
[2, 2, 3, 3, 3, 3]
2.67
b (minimal)
• {1, 3, 4, 5, 6, 11}
6
[2, 2, 3, 3, 3, 3]
2.67
c (minimal)
• {1, 3, 4, 6, 10, 11}
6
[2, 2, 3, 3, 3, 4]
2.83
d (minimal)
• {1, 3, 4, 5, 7, 11}
6
[2, 2, 3, 3, 3, 3]
2.67
e (minimal)
• {1, 2, 4, 5, 6, 11}
6
[2, 2, 3, 3, 3, 3]
2.67
f (minimal)
• {1, 2, 4, 6, 10, 11}
6
[2, 2, 3, 3, 3, 4]
2.83
g (minimal)
• {1, 2, 4, 5, 7, 11}
6
[2, 2, 3, 3, 3, 3]
2.67
h (minimal)
• {1, 2, 4, 6, 9, 11}
6
[2, 2, 3, 3, 3, 3]
2.67
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
1
0
0
2.6
6
0
8
7
2.76
7
0
0
55
2.95
8
0
0
92
3.09
9
0
0
82
3.19
10
0
0
40
3.27
11
0
0
10
3.31
12
0
0
1
3.33
Total
1
8
287
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],[0,5,6,2],[0,1,6,7],[0,8,9,4],[0,3,9,5],[1,4,6,6],[1,5,5,2],[2,9,8,8],[3,7,7,9],[3,8,7,4]]
PD code (use to draw this loop with SnapPy): [[20,7,1,8],[8,11,9,12],[12,19,13,20],[15,6,16,7],[1,16,2,17],[17,10,18,11],[9,18,10,19],[13,5,14,4],[14,3,15,4],[5,2,6,3]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (3,20,-4,-1)(12,1,-13,-2)(2,11,-3,-12)(13,4,-14,-5)(18,5,-19,-6)(15,8,-16,-9)(6,9,-7,-10)(19,14,-20,-15)(7,16,-8,-17)(10,17,-11,-18)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,12,-3)(-2,-12)(-4,13,1)(-5,18,-11,2,-13)(-6,-10,-18)(-7,-17,10)(-8,15,-20,3,11,17)(-9,6,-19,-15)(-14,19,5)(-16,7,9)(4,20,14)(8,16)
Loop annotated with half-edges
12^1_204 annotated with half-edges