$12^{2}_{238}$ - Minimal pinning sets
Pinning sets for 12^2_238 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_238 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, 4, 7, 11}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 2, 3, 7, 11}
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, 4, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,2,2],[0,1,1,3],[0,2,6,6],[0,7,7,5],[1,4,8,9],[3,9,7,3],[4,6,8,4],[5,7,9,9],[5,8,8,6]]
PD code (use to draw this multiloop with SnapPy): [[8,5,1,6],[6,9,7,20],[7,19,8,20],[4,18,5,19],[1,14,2,13],[9,13,10,12],[17,3,18,4],[14,3,15,2],[10,15,11,16],[16,11,17,12]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (9,8,-10,-1)(5,2,-6,-3)(7,18,-8,-19)(16,13,-17,-14)(17,6,-18,-7)(14,19,-15,-20)(1,10,-2,-11)(4,11,-5,-12)(12,3,-13,-4)(20,15,-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,-11,4,-13,16,-9)(-2,5,11)(-3,12,-5)(-4,-12)(-6,17,13,3)(-7,-19,14,-17)(-8,9,15,19)(-10,1)(-14,-20,-16)(-15,20)(-18,7)(2,10,8,18,6)
Multiloop annotated with half-edges
12^2_238 annotated with half-edges