$12^{2}_{224}$ - Minimal pinning sets
Pinning sets for 12^2_224 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_224 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, 6, 9}
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],[0,4,5,6],[0,6,7,0],[0,8,9,9],[1,9,9,8],[1,8,8,7],[1,7,7,2],[2,6,6,5],[3,5,5,4],[3,4,4,3]]
PD code (use to draw this multiloop with SnapPy): [[7,12,8,1],[6,20,7,13],[11,8,12,9],[1,16,2,15],[13,3,14,4],[17,5,18,6],[19,9,20,10],[10,18,11,19],[16,5,17,4],[2,14,3,15]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (13,12,-14,-1)(1,16,-2,-17)(17,2,-18,-3)(8,5,-9,-6)(19,6,-20,-7)(4,9,-5,-10)(15,10,-16,-11)(7,18,-8,-19)(3,20,-4,-13)(11,14,-12,-15)
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,-17,-3,-13)(-2,17)(-4,-10,15,-12,13)(-5,8,18,2,16,10)(-6,19,-8)(-7,-19)(-9,4,20,6)(-11,-15)(-14,11,-16,1)(-18,7,-20,3)(5,9)(12,14)
Multiloop annotated with half-edges
12^2_224 annotated with half-edges