$12^{2}_{244}$ - Minimal pinning sets
Pinning sets for 12^2_244 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_244 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, 3, 6, 9, 11}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 3, 6, 9, 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, 3, 4, 5, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,2],[0,1,3,3],[0,2,2,6],[0,7,8,8],[1,8,9,1],[3,9,7,7],[4,6,6,9],[4,9,5,4],[5,8,7,6]]
PD code (use to draw this multiloop with SnapPy): [[5,8,6,1],[4,20,5,9],[7,19,8,20],[6,19,7,18],[1,16,2,15],[9,3,10,4],[17,12,18,13],[16,12,17,11],[2,14,3,15],[10,14,11,13]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (9,8,-10,-1)(2,5,-3,-6)(11,16,-12,-17)(17,12,-18,-13)(13,10,-14,-11)(7,14,-8,-15)(15,6,-16,-7)(18,3,-19,-4)(4,19,-5,-20)(1,20,-2,-9)
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,-9)(-2,-6,15,-8,9)(-3,18,12,16,6)(-4,-20,1,-10,13,-18)(-5,2,20)(-7,-15)(-11,-17,-13)(-12,17)(-14,7,-16,11)(-19,4)(3,5,19)(8,14,10)
Multiloop annotated with half-edges
12^2_244 annotated with half-edges