$12^{1}_{74}$ - Minimal pinning sets
Pinning sets for 12^1_74 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_74 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, 5, 11}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 2, 3, 5, 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, 4, 4, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,6],[0,6,7,7],[0,7,8,9],[0,6,5,5],[1,4,4,1],[1,4,9,2],[2,8,3,2],[3,7,9,9],[3,8,8,6]]
PD code (use to draw this loop with SnapPy): [[13,20,14,1],[3,12,4,13],[19,8,20,9],[14,18,15,17],[1,10,2,11],[11,2,12,3],[4,10,5,9],[7,18,8,19],[15,7,16,6],[16,5,17,6]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (9,20,-10,-1)(14,3,-15,-4)(4,13,-5,-14)(17,6,-18,-7)(7,2,-8,-3)(19,8,-20,-9)(1,10,-2,-11)(15,12,-16,-13)(5,16,-6,-17)(11,18,-12,-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,-11,-19,-9)(-2,7,-18,11)(-3,14,-5,-17,-7)(-4,-14)(-6,17)(-8,19,-12,15,3)(-10,1)(-13,4,-15)(-16,5,13)(-20,9)(2,10,20,8)(6,16,12,18)
Loop annotated with half-edges
12^1_74 annotated with half-edges