$12^{1}_{18}$ - Minimal pinning sets
Pinning sets for 12^1_18 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_18 Pinning data
Pinning number of this loop: 4
Total number of pinning sets: 436
of which optimal: 1
of which minimal: 8
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.05986
on average over minimal pinning sets: 2.5625
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, 5, 7}
4
[2, 2, 2, 4]
2.50
a (minimal)
• {1, 2, 4, 7, 10}
5
[2, 2, 2, 3, 4]
2.60
b (minimal)
• {1, 2, 6, 7, 10}
5
[2, 2, 2, 3, 3]
2.40
c (minimal)
• {1, 2, 6, 7, 9}
5
[2, 2, 2, 3, 4]
2.60
d (minimal)
• {1, 2, 3, 6, 7}
5
[2, 2, 2, 3, 3]
2.40
e (minimal)
• {1, 2, 3, 7, 10}
5
[2, 2, 2, 3, 3]
2.40
f (minimal)
• {1, 2, 3, 7, 12}
5
[2, 2, 2, 3, 4]
2.60
g (minimal)
• {1, 2, 4, 7, 9, 12}
6
[2, 2, 2, 4, 4, 4]
3.00
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
6
8
2.64
6
0
1
56
2.85
7
0
0
111
3.0
8
0
0
123
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
1
7
428
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,3,4,5],[0,5,4,3],[0,2,1,0],[1,2,6,7],[1,7,8,2],[4,9,9,7],[4,6,8,5],[5,7,9,9],[6,8,8,6]]
PD code (use to draw this loop with SnapPy): [[20,11,1,12],[12,10,13,9],[19,2,20,3],[10,1,11,2],[13,19,14,18],[8,3,9,4],[14,6,15,5],[17,4,18,5],[7,16,8,17],[6,16,7,15]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (5,20,-6,-1)(14,1,-15,-2)(2,11,-3,-12)(6,9,-7,-10)(17,8,-18,-9)(12,3,-13,-4)(4,13,-5,-14)(15,10,-16,-11)(7,18,-8,-19)(16,19,-17,-20)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,14,-5)(-2,-12,-4,-14)(-3,12)(-6,-10,15,1)(-7,-19,16,10)(-8,17,19)(-9,6,20,-17)(-11,2,-15)(-13,4)(-16,-20,5,13,3,11)(-18,7,9)(8,18)
Loop annotated with half-edges
12^1_18 annotated with half-edges