$12^{1}_{363}$ - Minimal pinning sets
Pinning sets for 12^1_363 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_363 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 228
of which optimal: 2
of which minimal: 6
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.98391
on average over minimal pinning sets: 2.51111
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, 5, 9, 11}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 2, 4, 5, 11}
5
[2, 2, 2, 2, 3]
2.20
a (minimal)
• {1, 2, 3, 5, 7, 11}
6
[2, 2, 2, 2, 4, 4]
2.67
b (minimal)
• {1, 2, 3, 5, 6, 11}
6
[2, 2, 2, 2, 4, 4]
2.67
c (minimal)
• {1, 2, 5, 7, 10, 11}
6
[2, 2, 2, 2, 4, 4]
2.67
d (minimal)
• {1, 2, 5, 6, 10, 11}
6
[2, 2, 2, 2, 4, 4]
2.67
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
2
0
0
2.2
6
0
4
13
2.57
7
0
0
48
2.82
8
0
0
68
2.99
9
0
0
56
3.11
10
0
0
28
3.2
11
0
0
8
3.27
12
0
0
1
3.33
Total
2
4
222
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,6,6],[0,7,7,8],[0,8,8,4],[0,3,5,5],[1,4,4,9],[1,9,7,1],[2,6,9,2],[2,9,3,3],[5,8,7,6]]
PD code (use to draw this loop with SnapPy): [[20,15,1,16],[16,3,17,4],[12,19,13,20],[7,14,8,15],[1,8,2,9],[9,2,10,3],[17,5,18,4],[18,11,19,12],[13,6,14,7],[10,6,11,5]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (7,20,-8,-1)(14,1,-15,-2)(2,15,-3,-16)(3,8,-4,-9)(11,4,-12,-5)(19,6,-20,-7)(16,9,-17,-10)(5,12,-6,-13)(18,13,-19,-14)(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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,14,-19,-7)(-2,-16,-10,-18,-14)(-3,-9,16)(-4,11,17,9)(-5,-13,18,-11)(-6,19,13)(-8,3,15,1)(-12,5)(-15,2)(-17,10)(-20,7)(4,8,20,6,12)
Loop annotated with half-edges
12^1_363 annotated with half-edges