$12^{2}_{20}$ - Minimal pinning sets
Pinning sets for 12^2_20 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_20 Pinning data
Pinning number of this multiloop: 3
Total number of pinning sets: 512
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: 3.03436
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)
• {3, 5, 7}
3
[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
3
1
0
0
2.0
4
0
0
9
2.44
5
0
0
36
2.71
6
0
0
84
2.89
7
0
0
126
3.02
8
0
0
126
3.11
9
0
0
84
3.19
10
0
0
36
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
1
0
511
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 3, 4, 5, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,4,5],[0,5,6,7],[0,8,9,9],[0,5,1,1],[1,4,9,2],[2,8,7,7],[2,6,6,8],[3,7,6,9],[3,8,5,3]]
PD code (use to draw this multiloop with SnapPy): [[3,14,4,1],[2,7,3,8],[10,13,11,14],[4,15,5,20],[1,9,2,8],[9,6,10,7],[16,12,17,13],[11,17,12,18],[15,18,16,19],[5,19,6,20]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (7,4,-8,-5)(14,5,-1,-6)(6,13,-7,-14)(17,10,-18,-11)(8,11,-9,-12)(1,12,-2,-13)(9,18,-10,-19)(16,19,-17,-20)(3,20,-4,-15)(15,2,-16,-3)
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,-13,6)(-2,15,-4,7,13)(-3,-15)(-5,14,-7)(-6,-14)(-8,-12,1,5)(-9,-19,16,2,12)(-10,17,19)(-11,8,4,20,-17)(-16,-20,3)(-18,9,11)(10,18)
Multiloop annotated with half-edges
12^2_20 annotated with half-edges