$12^{3}_{18}$ - Minimal pinning sets
Pinning sets for 12^3_18 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^3_18 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)
• {2, 4, 6, 7, 12}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {2, 4, 5, 7, 12}
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],[0,4,4,5],[0,5,6,6],[0,6,7,8],[0,9,1,1],[1,9,9,2],[2,7,3,2],[3,6,8,8],[3,7,7,9],[4,8,5,5]]
PD code (use to draw this multiloop with SnapPy): [[3,10,4,1],[2,14,3,11],[9,20,10,15],[4,8,5,7],[1,12,2,11],[13,15,14,16],[19,8,20,9],[5,19,6,18],[6,17,7,18],[12,17,13,16]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (5,2,-6,-3)(17,6,-18,-7)(1,8,-2,-9)(10,11,-1,-12)(12,9,-13,-10)(16,3,-17,-4)(7,18,-8,-19)(14,19,-11,-20)(20,13,-15,-14)(4,15,-5,-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,-9,12)(-2,5,15,13,9)(-3,16,-5)(-4,-16)(-6,17,3)(-7,-19,14,-15,4,-17)(-8,1,11,19)(-10,-12)(-11,10,-13,20)(-14,-20)(-18,7)(2,8,18,6)
Multiloop annotated with half-edges
12^3_18 annotated with half-edges