$12^{1}_{327}$ - Minimal pinning sets
Pinning sets for 12^1_327 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_327 Pinning data
Pinning number of this loop: 4
Total number of pinning sets: 636
of which optimal: 4
of which minimal: 6
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.12577
on average over minimal pinning sets: 2.68333
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)
• {1, 2, 5, 11}
4
[2, 2, 3, 3]
2.50
B (optimal)
• {1, 5, 8, 11}
4
[2, 2, 3, 4]
2.75
C (optimal)
• {1, 7, 11, 12}
4
[2, 2, 3, 3]
2.50
D (optimal)
• {1, 4, 7, 11}
4
[2, 2, 3, 4]
2.75
a (minimal)
• {1, 3, 5, 7, 11}
5
[2, 2, 3, 3, 4]
2.80
b (minimal)
• {1, 5, 7, 10, 11}
5
[2, 2, 3, 3, 4]
2.80
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
4
0
0
2.62
5
0
2
30
2.84
6
0
0
99
2.99
7
0
0
166
3.1
8
0
0
170
3.18
9
0
0
110
3.23
10
0
0
44
3.28
11
0
0
10
3.31
12
0
0
1
3.33
Total
4
2
630
Other information about this loop Properties
Region degree sequence: [2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,6,6],[0,6,7,3],[0,2,8,4],[0,3,9,5],[1,4,7,6],[1,5,2,1],[2,5,9,8],[3,7,9,9],[4,8,8,7]]
PD code (use to draw this loop with SnapPy): [[20,5,1,6],[6,14,7,13],[8,19,9,20],[9,4,10,5],[1,10,2,11],[14,11,15,12],[7,12,8,13],[15,18,16,19],[16,3,17,4],[2,17,3,18]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (13,20,-14,-1)(6,1,-7,-2)(17,2,-18,-3)(18,7,-19,-8)(11,8,-12,-9)(4,9,-5,-10)(10,3,-11,-4)(19,14,-20,-15)(12,15,-13,-16)(5,16,-6,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,6,16,-13)(-2,17,-6)(-3,10,-5,-17)(-4,-10)(-7,18,2)(-8,11,3,-18)(-9,4,-11)(-12,-16,5,9)(-14,19,7,1)(-15,12,8,-19)(-20,13,15)(14,20)
Loop annotated with half-edges
12^1_327 annotated with half-edges