$12^{2}_{260}$ - Minimal pinning sets
Pinning sets for 12^2_260 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_260 Pinning data
Pinning number of this multiloop: 6
Total number of pinning sets: 96
of which optimal: 2
of which minimal: 2
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.91189
on average over minimal pinning sets: 2.16667
on average over optimal pinning sets: 2.16667
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 6, 8, 9}
6
[2, 2, 2, 2, 2, 3]
2.17
B (optimal)
• {1, 2, 3, 6, 9, 10}
6
[2, 2, 2, 2, 2, 3]
2.17
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
6
2
0
0
2.17
7
0
0
11
2.52
8
0
0
25
2.78
9
0
0
30
2.98
10
0
0
20
3.13
11
0
0
7
3.25
12
0
0
1
3.33
Total
2
0
94
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 2, 3, 3, 3, 4, 5, 6, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,4,4,2],[0,1,4,5],[0,5,6,0],[1,7,2,1],[2,8,8,3],[3,9,7,7],[4,6,6,9],[5,9,9,5],[6,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[14,20,1,15],[15,12,16,11],[13,10,14,11],[19,1,20,2],[12,17,13,16],[9,2,10,3],[18,6,19,7],[17,6,18,5],[3,8,4,9],[7,4,8,5]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (7,14,-8,-1)(5,2,-6,-3)(3,20,-4,-15)(15,4,-16,-5)(1,6,-2,-7)(11,8,-12,-9)(9,18,-10,-19)(16,13,-17,-14)(19,10,-20,-11)(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,-7)(-2,5,-16,-14,7)(-3,-15,-5)(-4,15)(-6,1,-8,11,-20,3)(-9,-19,-11)(-10,19)(-12,-18,9)(-13,16,4,20,10,18)(-17,12,8,14)(2,6)(13,17)
Multiloop annotated with half-edges
12^2_260 annotated with half-edges