$12^{2}_{89}$ - Minimal pinning sets
Pinning sets for 12^2_89 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_89 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 256
of which optimal: 1
of which minimal: 1
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.96564
on average over minimal pinning sets: 2.0
on average over optimal pinning sets: 2.0
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 8, 11}
4
[2, 2, 2, 2]
2.00
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
2.0
5
0
0
8
2.4
6
0
0
28
2.67
7
0
0
56
2.86
8
0
0
70
3.0
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
1
0
255
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 4, 4, 5, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,4,5],[0,6,6,3],[0,2,7,8],[0,9,1,1],[1,9,9,6],[2,5,7,2],[3,6,8,8],[3,7,7,9],[4,8,5,5]]
PD code (use to draw this multiloop with SnapPy): [[3,16,4,1],[2,20,3,17],[15,8,16,9],[4,8,5,7],[1,18,2,17],[13,19,14,20],[9,14,10,15],[5,10,6,11],[11,6,12,7],[18,12,19,13]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (9,4,-10,-5)(5,2,-6,-3)(13,6,-14,-7)(3,8,-4,-9)(19,10,-20,-11)(7,12,-8,-13)(1,14,-2,-15)(11,20,-12,-17)(16,17,-1,-18)(18,15,-19,-16)
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,-15,18)(-2,5,-10,19,15)(-3,-9,-5)(-4,9)(-6,13,-8,3)(-7,-13)(-11,-17,16,-19)(-12,7,-14,1,17)(-16,-18)(-20,11)(2,14,6)(4,8,12,20,10)
Multiloop annotated with half-edges
12^2_89 annotated with half-edges