$12^{2}_{234}$ - Minimal pinning sets
Pinning sets for 12^2_234 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_234 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 128
of which optimal: 1
of which minimal: 1
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.90623
on average over minimal pinning sets: 2.0
on average over optimal pinning sets: 2.0
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 5, 7, 11}
5
[2, 2, 2, 2, 2]
2.00
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
1
0
0
2.0
6
0
0
7
2.38
7
0
0
21
2.65
8
0
0
35
2.86
9
0
0
35
3.02
10
0
0
21
3.14
11
0
0
7
3.25
12
0
0
1
3.33
Total
1
0
127
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 2, 3, 4, 4, 4, 5, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,2,0],[0,1,5,6],[0,7,7,8],[1,8,8,5],[2,4,9,9],[2,9,7,7],[3,6,6,3],[3,9,4,4],[5,8,6,5]]
PD code (use to draw this multiloop with SnapPy): [[16,7,1,8],[8,15,9,16],[9,6,10,7],[1,12,2,13],[14,20,15,17],[5,19,6,20],[10,4,11,3],[11,2,12,3],[13,18,14,17],[18,4,19,5]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (9,16,-10,-1)(1,8,-2,-9)(17,2,-18,-3)(7,4,-8,-5)(14,5,-15,-6)(15,10,-16,-11)(6,13,-7,-14)(3,18,-4,-19)(12,19,-13,-20)(20,11,-17,-12)
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)(-2,17,11,-16,9)(-3,-19,12,-17)(-4,7,13,19)(-5,14,-7)(-6,-14)(-8,1,-10,15,5)(-11,20,-13,6,-15)(-12,-20)(-18,3)(2,8,4,18)(10,16)
Multiloop annotated with half-edges
12^2_234 annotated with half-edges