$12^{1}_{330}$ - Minimal pinning sets
Pinning sets for 12^1_330 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_330 Pinning data
Pinning number of this loop: 4
Total number of pinning sets: 384
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: 3.03466
on average over minimal pinning sets: 2.25
on average over optimal pinning sets: 2.25
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 5, 11}
4
[2, 2, 2, 3]
2.25
B (optimal)
• {1, 3, 7, 11}
4
[2, 2, 2, 3]
2.25
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
2
0
0
2.25
5
0
0
15
2.59
6
0
0
49
2.81
7
0
0
91
2.97
8
0
0
105
3.08
9
0
0
77
3.17
10
0
0
35
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
2
0
382
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 4, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,6,2],[0,1,6,7],[0,7,7,8],[0,8,5,5],[1,4,4,9],[1,9,9,2],[2,8,3,3],[3,7,9,4],[5,8,6,6]]
PD code (use to draw this loop with SnapPy): [[20,7,1,8],[8,3,9,4],[4,19,5,20],[15,6,16,7],[1,13,2,12],[2,11,3,12],[9,18,10,19],[5,14,6,15],[16,14,17,13],[17,10,18,11]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (6,1,-7,-2)(13,4,-14,-5)(18,5,-19,-6)(15,8,-16,-9)(20,9,-1,-10)(10,19,-11,-20)(11,14,-12,-15)(3,12,-4,-13)(7,16,-8,-17)(2,17,-3,-18)
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,-19,10)(-2,-18,-6)(-3,-13,-5,18)(-4,13)(-7,-17,2)(-8,15,-12,3,17)(-9,20,-11,-15)(-10,-20)(-14,11,19,5)(-16,7,1,9)(4,12,14)(8,16)
Loop annotated with half-edges
12^1_330 annotated with half-edges