$12^{2}_{19}$ - Minimal pinning sets
Pinning sets for 12^2_19 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_19 Pinning data
Pinning number of this multiloop: 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, 6, 8}
4
[2, 2, 2, 3]
2.25
a (minimal)
• {1, 3, 6, 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 multiloop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 3, 4, 5, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,3,3,4],[0,5,6,0],[0,4,1,1],[1,3,6,5],[2,4,7,8],[2,8,9,4],[5,9,9,8],[5,7,9,6],[6,8,7,7]]
PD code (use to draw this multiloop with SnapPy): [[3,10,4,1],[2,20,3,11],[9,4,10,5],[1,12,2,11],[12,19,13,20],[5,13,6,14],[18,8,19,9],[6,16,7,17],[14,17,15,18],[15,7,16,8]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (7,2,-8,-3)(16,5,-17,-6)(19,6,-20,-7)(1,8,-2,-9)(3,14,-4,-15)(4,17,-5,-18)(15,18,-16,-19)(13,20,-14,-11)(10,11,-1,-12)(12,9,-13,-10)
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,-9,12)(-2,7,-20,13,9)(-3,-15,-19,-7)(-4,-18,15)(-5,16,18)(-6,19,-16)(-8,1,11,-14,3)(-10,-12)(-11,10,-13)(-17,4,14,20,6)(2,8)(5,17)
Multiloop annotated with half-edges
12^2_19 annotated with half-edges