$12^{3}_{55}$ - Minimal pinning sets
Pinning sets for 12^3_55 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^3_55 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, 3, 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, 4, 4, 4, 4, 4, 4, 6]
Minimal region degree: 2
Is multisimple: Yes
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,5,0],[0,5,5,6],[0,6,6,4],[1,3,7,8],[1,8,2,2],[2,7,3,3],[4,6,9,9],[4,9,9,5],[7,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[10,14,1,11],[11,9,12,10],[13,3,14,4],[1,15,2,20],[8,19,9,20],[12,5,13,4],[2,15,3,16],[16,7,17,8],[18,5,19,6],[6,17,7,18]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (18,1,-19,-2)(2,17,-3,-18)(14,3,-15,-4)(12,5,-13,-6)(20,7,-17,-8)(8,19,-9,-20)(4,13,-5,-14)(6,15,-7,-16)(16,9,-11,-10)(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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,18,-3,14,-5,12)(-2,-18)(-4,-14)(-6,-16,-10,-12)(-7,20,-9,16)(-8,-20)(-11,10)(-13,4,-15,6)(-17,2,-19,8)(1,11,9,19)(3,17,7,15)(5,13)
Multiloop annotated with half-edges
12^3_55 annotated with half-edges