$12^{1}_{92}$ - Minimal pinning sets
Pinning sets for 12^1_92 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_92 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 232
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: 3.03951
on average over minimal pinning sets: 2.53889
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, 3, 4, 9, 10}
5
[2, 2, 2, 3, 3]
2.40
a (minimal)
• {1, 2, 4, 9, 10, 11}
6
[2, 2, 2, 3, 3, 3]
2.50
b (minimal)
• {1, 2, 4, 8, 9, 11}
6
[2, 2, 2, 3, 3, 3]
2.50
c (minimal)
• {1, 2, 4, 6, 9, 10}
6
[2, 2, 2, 3, 3, 4]
2.67
d (minimal)
• {1, 2, 4, 7, 9, 10}
6
[2, 2, 2, 3, 3, 4]
2.67
e (minimal)
• {1, 3, 4, 8, 9, 11}
6
[2, 2, 2, 3, 3, 3]
2.50
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
2.62
7
0
0
42
2.85
8
0
0
70
3.02
9
0
0
64
3.14
10
0
0
33
3.23
11
0
0
9
3.29
12
0
0
1
3.33
Total
1
5
226
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: [[1,2,3,4],[0,4,5,6],[0,6,7,7],[0,7,8,9],[0,5,5,1],[1,4,4,6],[1,5,9,2],[2,8,3,2],[3,7,9,9],[3,8,8,6]]
PD code (use to draw this loop with SnapPy): [[13,20,14,1],[12,9,13,10],[4,19,5,20],[14,18,15,17],[1,11,2,10],[2,11,3,12],[3,8,4,9],[18,5,19,6],[15,6,16,7],[7,16,8,17]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (15,2,-16,-3)(12,3,-13,-4)(20,5,-1,-6)(9,6,-10,-7)(7,18,-8,-19)(19,8,-20,-9)(4,13,-5,-14)(14,11,-15,-12)(1,16,-2,-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,-17,10,6)(-2,15,11,17)(-3,12,-15)(-4,-14,-12)(-5,20,8,18,-11,14)(-6,9,-20)(-7,-19,-9)(-8,19)(-10,-18,7)(-13,4)(-16,1,5,13,3)(2,16)
Loop annotated with half-edges
12^1_92 annotated with half-edges