$12^{2}_{40}$ - Minimal pinning sets
Pinning sets for 12^2_40 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_40 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, 9, 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, 4, 4, 4, 4, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,4,4,5],[0,6,6,0],[0,6,7,4],[1,3,8,1],[1,9,9,7],[2,7,3,2],[3,6,5,8],[4,7,9,9],[5,8,8,5]]
PD code (use to draw this multiloop with SnapPy): [[14,20,1,15],[15,7,16,8],[19,13,20,14],[1,18,2,17],[6,16,7,17],[8,4,9,3],[12,18,13,19],[2,12,3,11],[5,10,6,11],[4,10,5,9]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (14,5,-1,-6)(15,4,-16,-5)(6,1,-7,-2)(2,7,-3,-8)(8,13,-9,-14)(19,10,-20,-11)(11,20,-12,-15)(3,16,-4,-17)(17,12,-18,-13)(9,18,-10,-19)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,6)(-2,-8,-14,-6)(-3,-17,-13,8)(-4,15,-12,17)(-5,14,-9,-19,-11,-15)(-7,2)(-10,19)(-16,3,7,1,5)(-18,9,13)(-20,11)(4,16)(10,18,12,20)
Multiloop annotated with half-edges
12^2_40 annotated with half-edges