$12^{3}_{81}$ - Minimal pinning sets
Pinning sets for 12^3_81 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^3_81 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, 2, 3, 4}
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, 3, 3, 5, 5, 5, 5]
Minimal region degree: 2
Is multisimple: Yes
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,3,4,5],[0,6,6,0],[0,7,8,1],[1,9,9,5],[1,4,7,6],[2,5,7,2],[3,6,5,8],[3,7,9,9],[4,8,8,4]]
PD code (use to draw this multiloop with SnapPy): [[8,16,1,9],[9,17,10,20],[15,7,16,8],[1,18,2,17],[10,4,11,5],[5,19,6,20],[6,14,7,15],[18,14,19,13],[2,13,3,12],[3,11,4,12]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (15,2,-16,-3)(12,5,-13,-6)(1,16,-2,-9)(9,8,-10,-1)(10,17,-11,-18)(4,11,-5,-12)(6,13,-7,-14)(3,18,-4,-19)(19,14,-20,-15)(20,7,-17,-8)
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,-9)(-2,15,-20,-8,9)(-3,-19,-15)(-4,-12,-6,-14,19)(-5,12)(-7,20,14)(-10,-18,3,-16,1)(-11,4,18)(-13,6)(-17,10,8)(2,16)(5,11,17,7,13)
Multiloop annotated with half-edges
12^3_81 annotated with half-edges