$12^{2}_{309}$ - Minimal pinning sets
Pinning sets for 12^2_309 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_309 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 368
of which optimal: 1
of which minimal: 4
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.04152
on average over minimal pinning sets: 2.4625
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, 2, 5, 7}
4
[2, 2, 2, 3]
2.25
a (minimal)
• {1, 2, 6, 7, 12}
5
[2, 2, 2, 3, 4]
2.60
b (minimal)
• {1, 2, 6, 7, 10}
5
[2, 2, 2, 3, 3]
2.40
c (minimal)
• {1, 2, 3, 6, 7}
5
[2, 2, 2, 3, 4]
2.60
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
3
8
2.56
6
0
0
43
2.79
7
0
0
87
2.96
8
0
0
104
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
1
3
364
Other information about this multiloop 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],[0,5,6,7],[0,7,7,8],[0,8,8,9],[0,9,5,5],[1,4,4,6],[1,5,9,7],[1,6,2,2],[2,9,3,3],[3,8,6,4]]
PD code (use to draw this multiloop with SnapPy): [[6,20,1,7],[7,3,8,4],[17,5,18,6],[19,10,20,11],[1,13,2,14],[14,2,15,3],[8,15,9,16],[4,16,5,17],[18,12,19,11],[12,9,13,10]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (7,6,-8,-1)(14,1,-15,-2)(10,3,-11,-4)(19,16,-20,-17)(12,17,-13,-18)(18,11,-19,-12)(5,20,-6,-7)(15,8,-16,-9)(2,9,-3,-10)(4,13,-5,-14)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,14,-5,-7)(-2,-10,-4,-14)(-3,10)(-6,7)(-8,15,1)(-9,2,-15)(-11,18,-13,4)(-12,-18)(-16,19,11,3,9)(-17,12,-19)(-20,5,13,17)(6,20,16,8)
Multiloop annotated with half-edges
12^2_309 annotated with half-edges