$12^{1}_{355}$ - Minimal pinning sets
Pinning sets for 12^1_355 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_355 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 196
of which optimal: 1
of which minimal: 5
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.98956
on average over minimal pinning sets: 2.45905
on average over optimal pinning sets: 2.2
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 4, 11}
5
[2, 2, 2, 2, 3]
2.20
a (minimal)
• {1, 2, 4, 6, 9, 11}
6
[2, 2, 2, 2, 3, 4]
2.50
b (minimal)
• {1, 2, 4, 5, 10, 11}
6
[2, 2, 2, 2, 3, 4]
2.50
c (minimal)
• {1, 2, 4, 6, 10, 11}
6
[2, 2, 2, 2, 4, 4]
2.67
d (minimal)
• {1, 2, 4, 5, 8, 9, 11}
7
[2, 2, 2, 2, 3, 3, 3]
2.43
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
2.2
6
0
3
7
2.53
7
0
1
34
2.77
8
0
0
59
2.96
9
0
0
54
3.1
10
0
0
28
3.2
11
0
0
8
3.27
12
0
0
1
3.33
Total
1
4
191
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 5, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,5],[0,6,7,3],[0,2,8,9],[0,9,9,1],[1,6,6,1],[2,5,5,7],[2,6,8,8],[3,7,7,9],[3,8,4,4]]
PD code (use to draw this loop with SnapPy): [[15,20,16,1],[14,7,15,8],[19,4,20,5],[16,4,17,3],[1,9,2,8],[6,13,7,14],[5,13,6,12],[18,11,19,12],[17,11,18,10],[2,9,3,10]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (1,14,-2,-15)(17,2,-18,-3)(12,5,-13,-6)(6,19,-7,-20)(20,7,-1,-8)(8,15,-9,-16)(16,9,-17,-10)(10,3,-11,-4)(4,11,-5,-12)(13,18,-14,-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,-15,8)(-2,17,9,15)(-3,10,-17)(-4,-12,-6,-20,-8,-16,-10)(-5,12)(-7,20)(-9,16)(-11,4)(-13,-19,6)(-14,1,7,19)(-18,13,5,11,3)(2,14,18)
Loop annotated with half-edges
12^1_355 annotated with half-edges