$12^{5}_{1}$ - Minimal pinning sets
Pinning sets for 12^5_1 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^5_1 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, 3, 5, 7, 12}
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, 4, 4, 4, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: Yes
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,5,0],[0,5,5,6],[0,7,4,4],[1,3,3,8],[1,9,2,2],[2,9,9,7],[3,6,8,8],[4,7,7,9],[5,8,6,6]]
PD code (use to draw this multiloop with SnapPy): [[4,8,1,5],[5,3,6,4],[7,12,8,9],[1,13,2,16],[2,15,3,16],[6,10,7,9],[11,20,12,17],[13,20,14,19],[14,18,15,19],[10,18,11,17]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (18,1,-19,-2)(2,17,-3,-18)(20,15,-17,-16)(16,19,-13,-20)(4,5,-1,-6)(6,3,-7,-4)(8,9,-5,-10)(10,7,-11,-8)(14,11,-15,-12)(12,13,-9,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,18,-3,6)(-2,-18)(-4,-6)(-5,4,-7,10)(-8,-10)(-9,8,-11,14)(-12,-14)(-13,12,-15,20)(-16,-20)(-17,2,-19,16)(1,5,9,13,19)(3,17,15,11,7)
Multiloop annotated with half-edges
12^5_1 annotated with half-edges