$12^{2}_{68}$ - Minimal pinning sets
Pinning sets for 12^2_68 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_68 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 320
of which optimal: 1
of which minimal: 2
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.03463
on average over minimal pinning sets: 2.325
on average over optimal pinning sets: 2.25
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 5, 7}
4
[2, 2, 2, 3]
2.25
a (minimal)
• {1, 2, 4, 5, 7}
5
[2, 2, 2, 3, 3]
2.40
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
2.25
5
0
1
8
2.56
6
0
0
34
2.77
7
0
0
71
2.94
8
0
0
90
3.06
9
0
0
71
3.16
10
0
0
34
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
1
1
318
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 3, 3, 5, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,3,2,0],[0,1,4,5],[0,6,7,1],[2,8,5,5],[2,4,4,6],[3,5,9,7],[3,6,9,8],[4,7,9,9],[6,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[14,20,1,15],[15,13,16,14],[16,19,17,20],[1,12,2,13],[18,8,19,9],[17,8,18,7],[11,6,12,7],[2,6,3,5],[9,5,10,4],[10,3,11,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (8,1,-9,-2)(2,7,-3,-8)(3,14,-4,-1)(9,4,-10,-5)(5,20,-6,-15)(15,6,-16,-7)(13,10,-14,-11)(18,11,-19,-12)(19,16,-20,-17)(12,17,-13,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,8,-3)(-2,-8)(-4,9,1)(-5,-15,-7,2,-9)(-6,15)(-10,13,17,-20,5)(-11,18,-13)(-12,-18)(-14,3,7,-16,19,11)(-17,12,-19)(4,14,10)(6,20,16)
Multiloop annotated with half-edges
12^2_68 annotated with half-edges