$12^{3}_{5}$ - Minimal pinning sets
Pinning sets for 12^3_5 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^3_5 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 256
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: 2.96564
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, 6, 9}
4
[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
4
1
0
0
2.0
5
0
0
8
2.4
6
0
0
28
2.67
7
0
0
56
2.86
8
0
0
70
3.0
9
0
0
56
3.11
10
0
0
28
3.2
11
0
0
8
3.27
12
0
0
1
3.33
Total
1
0
255
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,4,4,5],[0,6,6,7],[0,8,4,0],[1,3,5,1],[1,4,8,6],[2,5,9,2],[2,9,9,8],[3,7,9,5],[6,8,7,7]]
PD code (use to draw this multiloop with SnapPy): [[12,16,1,13],[13,8,14,7],[11,20,12,17],[15,1,16,2],[8,15,9,14],[9,6,10,7],[17,10,18,11],[4,19,5,20],[2,5,3,6],[18,3,19,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (16,1,-13,-2)(12,3,-1,-4)(7,4,-8,-5)(5,20,-6,-17)(15,8,-16,-9)(18,11,-19,-12)(10,19,-11,-20)(17,6,-18,-7)(9,14,-10,-15)(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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,16,8,4)(-2,-14,9,-16)(-3,12,-19,10,14)(-4,7,-18,-12)(-5,-17,-7)(-6,17)(-8,15,-10,-20,5)(-9,-15)(-11,18,6,20)(-13,2)(1,3,13)(11,19)
Multiloop annotated with half-edges
12^3_5 annotated with half-edges