$12^{2}_{124}$ - Minimal pinning sets
Pinning sets for 12^2_124 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_124 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, 7, 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, 3, 4, 4, 4, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,4,4,2],[0,1,5,0],[0,6,7,4],[1,3,5,1],[2,4,8,9],[3,9,7,7],[3,6,6,8],[5,7,9,9],[5,8,8,6]]
PD code (use to draw this multiloop with SnapPy): [[7,16,8,1],[6,9,7,10],[15,8,16,9],[1,4,2,5],[10,5,11,6],[11,14,12,15],[3,20,4,17],[2,20,3,19],[13,18,14,19],[12,18,13,17]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (14,1,-15,-2)(7,2,-8,-3)(12,5,-13,-6)(3,6,-4,-7)(8,11,-9,-12)(4,13,-5,-14)(18,9,-19,-10)(10,19,-11,-20)(20,15,-17,-16)(16,17,-1,-18)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,14,-5,12,-9,18)(-2,7,-4,-14)(-3,-7)(-6,3,-8,-12)(-10,-20,-16,-18)(-11,8,2,-15,20)(-13,4,6)(-17,16)(-19,10)(1,17,15)(5,13)(9,11,19)
Multiloop annotated with half-edges
12^2_124 annotated with half-edges