$12^{2}_{242}$ - Minimal pinning sets
Pinning sets for 12^2_242 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_242 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, 2, 3, 4, 8}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 2, 3, 4, 6}
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, 5, 5, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,4,0],[0,5,6,3],[0,2,7,8],[1,8,5,1],[2,4,6,6],[2,5,5,9],[3,9,9,8],[3,7,9,4],[6,8,7,7]]
PD code (use to draw this multiloop with SnapPy): [[8,20,1,9],[9,7,10,8],[19,16,20,17],[1,16,2,15],[6,10,7,11],[17,6,18,5],[18,4,19,5],[2,13,3,12],[14,11,15,12],[3,13,4,14]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (9,8,-10,-1)(1,20,-2,-9)(12,3,-13,-4)(17,14,-18,-15)(15,6,-16,-7)(7,16,-8,-17)(18,5,-19,-6)(10,19,-11,-20)(4,11,-5,-12)(2,13,-3,-14)
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,-14,17,-8,9)(-3,12,-5,18,14)(-4,-12)(-6,15,-18)(-7,-17,-15)(-10,-20,1)(-11,4,-13,2,20)(-16,7)(-19,10,8,16,6)(3,13)(5,11,19)
Multiloop annotated with half-edges
12^2_242 annotated with half-edges