$12^{3}_{46}$ - Minimal pinning sets
Pinning sets for 12^3_46 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^3_46 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, 4, 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, 3, 4, 4, 4, 4, 5, 6]
Minimal region degree: 2
Is multisimple: Yes
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,4,0],[0,4,5,5],[0,5,6,7],[1,8,2,1],[2,9,3,2],[3,9,9,7],[3,6,8,8],[4,7,7,9],[5,8,6,6]]
PD code (use to draw this multiloop with SnapPy): [[8,16,1,9],[9,7,10,8],[15,5,16,6],[1,14,2,13],[6,10,7,11],[4,14,5,15],[2,17,3,20],[12,19,13,20],[11,19,12,18],[3,17,4,18]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (13,2,-14,-3)(15,4,-16,-5)(19,6,-20,-7)(1,20,-2,-13)(5,14,-6,-15)(3,16,-4,-17)(10,17,-11,-18)(18,11,-19,-12)(12,7,-9,-8)(8,9,-1,-10)
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,-13,-3,-17,10)(-2,13)(-4,15,-6,19,11,17)(-5,-15)(-7,12,-19)(-8,-10,-18,-12)(-9,8)(-11,18)(-14,5,-16,3)(-20,1,9,7)(2,20,6,14)(4,16)
Multiloop annotated with half-edges
12^3_46 annotated with half-edges