$12^{2}_{23}$ - Minimal pinning sets
Pinning sets for 12^2_23 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_23 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, 4, 6, 9}
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],[0,6,6,7],[0,8,8,5],[0,5,1,1],[1,4,3,6],[2,5,9,2],[2,9,9,8],[3,7,9,3],[6,8,7,7]]
PD code (use to draw this multiloop with SnapPy): [[3,16,4,1],[2,11,3,12],[15,20,16,17],[4,9,5,10],[1,13,2,12],[13,10,14,11],[17,14,18,15],[19,6,20,7],[8,5,9,6],[18,8,19,7]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (6,3,-7,-4)(17,4,-18,-5)(5,20,-6,-17)(2,7,-3,-8)(11,8,-12,-9)(16,9,-1,-10)(10,15,-11,-16)(19,12,-20,-13)(1,14,-2,-15)(13,18,-14,-19)
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,-15,10)(-2,-8,11,15)(-3,6,20,12,8)(-4,17,-6)(-5,-17)(-7,2,14,18,4)(-9,16,-11)(-10,-16)(-12,19,-14,1,9)(-13,-19)(-18,13,-20,5)(3,7)
Multiloop annotated with half-edges
12^2_23 annotated with half-edges