$12^{1}_{80}$ - Minimal pinning sets
Pinning sets for 12^1_80 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_80 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 192
of which optimal: 2
of which minimal: 2
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.96906
on average over minimal pinning sets: 2.2
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, 2, 4, 7, 11}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 2, 6, 7, 11}
5
[2, 2, 2, 2, 3]
2.20
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
2
0
0
2.2
6
0
0
13
2.54
7
0
0
36
2.78
8
0
0
55
2.95
9
0
0
50
3.09
10
0
0
27
3.19
11
0
0
8
3.27
12
0
0
1
3.33
Total
2
0
190
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 3, 4, 5, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,3,2,0],[0,1,4,5],[0,6,6,1],[2,7,8,8],[2,9,7,6],[3,5,7,3],[4,6,5,9],[4,9,9,4],[5,8,8,7]]
PD code (use to draw this loop with SnapPy): [[11,20,12,1],[19,10,20,11],[12,10,13,9],[1,18,2,19],[13,7,14,6],[8,3,9,4],[17,2,18,3],[7,17,8,16],[14,5,15,6],[4,15,5,16]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (15,20,-16,-1)(6,1,-7,-2)(12,3,-13,-4)(4,11,-5,-12)(7,10,-8,-11)(17,8,-18,-9)(2,13,-3,-14)(14,5,-15,-6)(19,16,-20,-17)(9,18,-10,-19)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,6,-15)(-2,-14,-6)(-3,12,-5,14)(-4,-12)(-7,-11,4,-13,2)(-8,17,-20,15,5,11)(-9,-19,-17)(-10,7,1,-16,19)(-18,9)(3,13)(8,10,18)(16,20)
Loop annotated with half-edges
12^1_80 annotated with half-edges