$12^{2}_{241}$ - Minimal pinning sets
Pinning sets for 12^2_241 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_241 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 192
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: 2.96906
on average over minimal pinning sets: 2.2
on average over optimal pinning sets: 2.2
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 4, 5, 7, 12}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 4, 5, 7, 10}
5
[2, 2, 2, 2, 3]
2.20
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
2
0
0
2.2
6
0
0
13
2.54
7
0
0
36
2.78
8
0
0
55
2.95
9
0
0
50
3.09
10
0
0
27
3.19
11
0
0
8
3.27
12
0
0
1
3.33
Total
2
0
190
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: [[1,2,2,3],[0,4,5,6],[0,7,7,0],[0,8,9,4],[1,3,5,5],[1,4,4,6],[1,5,9,7],[2,6,8,2],[3,7,9,9],[3,8,8,6]]
PD code (use to draw this multiloop with SnapPy): [[10,20,1,11],[11,14,12,15],[19,9,20,10],[1,4,2,5],[5,13,6,14],[12,6,13,7],[15,7,16,8],[8,18,9,19],[3,17,4,18],[2,17,3,16]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (9,2,-10,-3)(17,4,-18,-5)(20,5,-11,-6)(14,7,-15,-8)(3,18,-4,-19)(16,19,-17,-20)(11,10,-12,-1)(1,12,-2,-13)(6,13,-7,-14)(8,15,-9,-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,-13,6,-11)(-2,9,15,7,13)(-3,-19,16,-9)(-4,17,19)(-5,20,-17)(-6,-14,-8,-16,-20)(-7,14)(-10,11,5,-18,3)(-12,1)(-15,8)(2,12,10)(4,18)
Multiloop annotated with half-edges
12^2_241 annotated with half-edges