$12^{1}_{139}$ - Minimal pinning sets
Pinning sets for 12^1_139 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_139 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 244
of which optimal: 1
of which minimal: 7
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.0456
on average over minimal pinning sets: 2.58435
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, 2, 3, 7, 8}
5
[2, 2, 2, 3, 3]
2.40
a (minimal)
• {1, 3, 5, 7, 8, 11}
6
[2, 2, 2, 3, 3, 3]
2.50
b (minimal)
• {1, 3, 5, 7, 9, 11}
6
[2, 2, 2, 3, 3, 3]
2.50
c (minimal)
• {1, 3, 4, 7, 8, 11}
6
[2, 2, 2, 3, 3, 4]
2.67
d (minimal)
• {1, 2, 3, 6, 7, 9}
6
[2, 2, 2, 3, 3, 4]
2.67
e (minimal)
• {1, 2, 3, 5, 7, 9}
6
[2, 2, 2, 3, 3, 3]
2.50
f (minimal)
• {1, 3, 4, 6, 7, 9, 11}
7
[2, 2, 2, 3, 3, 4, 4]
2.86
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
2.4
6
0
5
7
2.63
7
0
1
43
2.85
8
0
0
75
3.02
9
0
0
68
3.15
10
0
0
34
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
1
6
237
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,3,4,2],[0,1,5,0],[0,6,7,1],[1,8,8,5],[2,4,8,6],[3,5,9,7],[3,6,9,9],[4,9,5,4],[6,8,7,7]]
PD code (use to draw this loop with SnapPy): [[7,20,8,1],[6,9,7,10],[19,8,20,9],[1,11,2,10],[5,16,6,17],[18,15,19,16],[11,15,12,14],[2,14,3,13],[17,4,18,5],[12,4,13,3]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (1,10,-2,-11)(15,2,-16,-3)(17,6,-18,-7)(4,7,-5,-8)(8,19,-9,-20)(11,20,-12,-1)(12,9,-13,-10)(16,13,-17,-14)(3,14,-4,-15)(5,18,-6,-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,-11)(-2,15,-4,-8,-20,11)(-3,-15)(-5,-19,8)(-6,17,13,9,19)(-7,4,14,-17)(-9,12,20)(-10,1,-12)(-13,16,2,10)(-14,3,-16)(-18,5,7)(6,18)
Loop annotated with half-edges
12^1_139 annotated with half-edges