$12^{1}_{332}$ - Minimal pinning sets
Pinning sets for 12^1_332 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_332 Pinning data
Pinning number of this loop: 4
Total number of pinning sets: 320
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: 3.03463
on average over minimal pinning sets: 2.325
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, 9}
4
[2, 2, 2, 3]
2.25
a (minimal)
• {1, 3, 5, 7, 10}
5
[2, 2, 2, 3, 3]
2.40
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
1
0
0
2.25
5
0
1
8
2.56
6
0
0
34
2.77
7
0
0
71
2.94
8
0
0
90
3.06
9
0
0
71
3.16
10
0
0
34
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
1
1
318
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,5,0],[0,5,6,3],[0,2,6,4],[1,3,6,7],[1,8,8,2],[2,9,4,3],[4,9,9,8],[5,7,9,5],[6,8,7,7]]
PD code (use to draw this loop with SnapPy): [[15,20,16,1],[19,14,20,15],[16,8,17,7],[1,7,2,6],[18,5,19,6],[13,8,14,9],[17,3,18,2],[11,4,12,5],[9,12,10,13],[3,10,4,11]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (8,1,-9,-2)(15,2,-16,-3)(3,18,-4,-19)(4,13,-5,-14)(10,5,-11,-6)(20,7,-1,-8)(6,9,-7,-10)(16,11,-17,-12)(12,17,-13,-18)(19,14,-20,-15)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,8)(-2,15,-20,-8)(-3,-19,-15)(-4,-14,19)(-5,10,-7,20,14)(-6,-10)(-9,6,-11,16,2)(-12,-18,3,-16)(-13,4,18)(-17,12)(1,7,9)(5,13,17,11)
Loop annotated with half-edges
12^1_332 annotated with half-edges