$12^{3}_{50}$ - Minimal pinning sets
Pinning sets for 12^3_50 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^3_50 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, 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, 3, 4, 4, 5, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,5,0],[0,5,5,6],[0,7,8,8],[1,9,9,5],[1,4,2,2],[2,9,7,7],[3,6,6,8],[3,7,9,3],[4,8,6,4]]
PD code (use to draw this multiloop with SnapPy): [[6,10,1,7],[7,5,8,6],[9,20,10,11],[1,18,2,17],[4,12,5,13],[8,12,9,11],[14,19,15,20],[18,15,19,16],[2,16,3,17],[13,3,14,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (14,1,-15,-2)(18,3,-19,-4)(4,17,-5,-18)(9,16,-10,-7)(6,7,-1,-8)(8,5,-9,-6)(15,10,-16,-11)(20,11,-17,-12)(12,19,-13,-20)(2,13,-3,-14)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,14,-3,18,-5,8)(-2,-14)(-4,-18)(-6,-8)(-7,6,-9)(-10,15,1,7)(-11,20,-13,2,-15)(-12,-20)(-16,9,5,17,11)(-17,4,-19,12)(3,13,19)(10,16)
Multiloop annotated with half-edges
12^3_50 annotated with half-edges