$12^{1}_{72}$ - Minimal pinning sets
Pinning sets for 12^1_72 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_72 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 160
of which optimal: 1
of which minimal: 2
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.97043
on average over minimal pinning sets: 2.26667
on average over optimal pinning sets: 2.2
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 4, 5, 6, 7}
5
[2, 2, 2, 2, 3]
2.20
a (minimal)
• {1, 2, 3, 5, 6, 7}
6
[2, 2, 2, 2, 3, 3]
2.33
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
2.2
6
0
1
7
2.5
7
0
0
26
2.74
8
0
0
45
2.92
9
0
0
45
3.07
10
0
0
26
3.18
11
0
0
8
3.27
12
0
0
1
3.33
Total
1
1
158
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 4, 4, 5, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,2],[0,3,3,0],[0,4,5,0],[1,5,4,1],[2,3,6,7],[2,7,8,3],[4,8,9,9],[4,9,8,5],[5,7,9,6],[6,8,7,6]]
PD code (use to draw this loop with SnapPy): [[9,20,10,1],[19,8,20,9],[10,2,11,1],[7,18,8,19],[2,18,3,17],[11,6,12,7],[3,15,4,14],[5,16,6,17],[12,16,13,15],[4,13,5,14]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (8,3,-9,-4)(16,5,-17,-6)(1,10,-2,-11)(11,20,-12,-1)(12,9,-13,-10)(2,13,-3,-14)(14,19,-15,-20)(4,15,-5,-16)(6,17,-7,-18)(18,7,-19,-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,-11)(-2,-14,-20,11)(-3,8,-19,14)(-4,-16,-6,-18,-8)(-5,16)(-7,18)(-9,12,20,-15,4)(-10,1,-12)(-13,2,10)(-17,6)(3,13,9)(5,15,19,7,17)
Loop annotated with half-edges
12^1_72 annotated with half-edges