$12^{2}_{230}$ - Minimal pinning sets
Pinning sets for 12^2_230 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_230 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, 3, 4, 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, 3, 4, 4, 5, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,4,4,5],[0,5,6,0],[0,7,8,8],[1,8,9,1],[1,6,6,2],[2,5,5,9],[3,9,9,8],[3,7,4,3],[4,7,7,6]]
PD code (use to draw this multiloop with SnapPy): [[5,10,6,1],[4,20,5,11],[9,6,10,7],[1,16,2,15],[11,3,12,4],[19,7,20,8],[8,18,9,19],[16,13,17,14],[2,14,3,15],[12,17,13,18]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (5,2,-6,-3)(20,3,-11,-4)(1,6,-2,-7)(12,9,-13,-10)(17,14,-18,-15)(8,15,-9,-16)(16,7,-17,-8)(13,18,-14,-19)(4,19,-5,-20)(10,11,-1,-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,-7,16,-9,12)(-2,5,19,-14,17,7)(-3,20,-5)(-4,-20)(-6,1,11,3)(-8,-16)(-10,-12)(-11,10,-13,-19,4)(-15,8,-17)(-18,13,9,15)(2,6)(14,18)
Multiloop annotated with half-edges
12^2_230 annotated with half-edges