$12^{1}_{124}$ - Minimal pinning sets
Pinning sets for 12^1_124 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_124 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, 3, 4, 5, 11}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 2, 4, 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, 3, 3, 4, 5, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,4,5],[0,6,7,7],[0,7,7,8],[0,5,1,1],[1,4,8,9],[2,9,9,8],[2,3,3,2],[3,6,9,5],[5,8,6,6]]
PD code (use to draw this loop with SnapPy): [[3,20,4,1],[11,2,12,3],[19,6,20,7],[4,18,5,17],[1,10,2,11],[12,10,13,9],[7,15,8,14],[5,18,6,19],[16,13,17,14],[8,15,9,16]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (9,20,-10,-1)(17,4,-18,-5)(7,10,-8,-11)(19,8,-20,-9)(14,11,-15,-12)(12,5,-13,-6)(6,13,-7,-14)(15,2,-16,-3)(3,16,-4,-17)(1,18,-2,-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,-19,-9)(-2,15,11,-8,19)(-3,-17,-5,12,-15)(-4,17)(-6,-14,-12)(-7,-11,14)(-10,7,13,5,-18,1)(-13,6)(-16,3)(-20,9)(2,18,4,16)(8,10,20)
Loop annotated with half-edges
12^1_124 annotated with half-edges