$12^{3}_{54}$ - Minimal pinning sets
Pinning sets for 12^3_54 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^3_54 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, 4, 6, 7}
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, 4, 4, 4, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: Yes
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,4,5,6],[0,6,7,8],[0,8,8,0],[1,8,7,7],[1,7,9,9],[1,9,9,2],[2,5,4,4],[2,4,3,3],[5,6,6,5]]
PD code (use to draw this multiloop with SnapPy): [[6,14,1,7],[7,15,8,20],[11,5,12,6],[13,1,14,2],[15,3,16,4],[8,17,9,18],[10,19,11,20],[4,16,5,17],[12,3,13,2],[9,19,10,18]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (6,7,-1,-8)(20,1,-15,-2)(14,3,-7,-4)(8,5,-9,-6)(4,9,-5,-10)(2,15,-3,-16)(11,16,-12,-17)(17,12,-18,-13)(13,18,-14,-19)(19,10,-20,-11)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,20,10,-5,8)(-2,-16,11,-20)(-3,14,18,12,16)(-4,-10,19,-14)(-6,-8)(-7,6,-9,4)(-11,-17,-13,-19)(-12,17)(-15,2)(-18,13)(1,7,3,15)(5,9)
Multiloop annotated with half-edges
12^3_54 annotated with half-edges