- Minimal pinning sets
Pinning sets for 12^1_298 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_298 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 208
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: 2.99175
on average over minimal pinning sets: 2.4
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, 2, 4, 8, 11}
5
[2, 2, 2, 2, 4]
2.40
a (minimal)
• {1, 2, 4, 6, 9, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
b (minimal)
• {1, 2, 4, 5, 9, 11}
6
[2, 2, 2, 2, 3, 4]
2.50
c (minimal)
• {1, 2, 4, 6, 10, 11}
6
[2, 2, 2, 2, 3, 4]
2.50
d (minimal)
• {1, 2, 4, 6, 7, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
e (minimal)
• {1, 2, 4, 7, 9, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
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
2.56
7
0
0
40
2.79
8
0
0
63
2.98
9
0
0
55
3.11
10
0
0
28
3.2
11
0
0
8
3.27
12
0
0
1
3.33
Total
1
5
202
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,6,2],[0,1,7,8],[0,8,4,4],[0,3,3,5],[1,4,9,9],[1,9,9,7],[2,6,8,8],[2,7,7,3],[5,6,6,5]]
PD code (use to draw this loop with SnapPy): [[20,3,1,4],[4,15,5,16],[16,19,17,20],[2,9,3,10],[1,9,2,8],[14,7,15,8],[5,13,6,12],[18,11,19,12],[17,11,18,10],[6,13,7,14]]
Permutation representation (action on half-edges):
Vertex permutation (18,1,-19,-2)(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)(2,13,-3,-14)(14,17,-15,-18)
Edge permutation (-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 (-1,18,-15,8)(-2,-14,-18)(-3,10,-17,14)(-4,-12,-6,-20,-8,-16,-10)(-5,12)(-7,20)(-9,16)(-11,4)(-13,2,-19,6)(1,7,19)(3,13,5,11)(9,15,17)
Loop annotated with half-edges
12^1_298 annotated with half-edges