$12^{2}_{317}$ - Minimal pinning sets
Pinning sets for 12^2_317 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_317 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 360
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: 3.04648
on average over minimal pinning sets: 2.52917
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, 3, 7}
4
[2, 2, 2, 3]
2.25
a (minimal)
• {1, 2, 3, 8, 10}
5
[2, 2, 2, 3, 4]
2.60
b (minimal)
• {1, 2, 3, 5, 8}
5
[2, 2, 2, 3, 4]
2.60
c (minimal)
• {1, 2, 3, 4, 8, 12}
6
[2, 2, 2, 3, 3, 4]
2.67
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
2
8
2.58
6
0
1
39
2.8
7
0
0
84
2.96
8
0
0
103
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
356
Other information about this multiloop 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],[0,4,4,0],[0,5,6,3],[0,2,7,8],[1,8,5,1],[2,4,8,9],[2,9,7,7],[3,6,6,9],[3,9,5,4],[5,8,7,6]]
PD code (use to draw this multiloop with SnapPy): [[16,5,1,6],[6,15,7,16],[4,20,5,17],[1,20,2,19],[14,7,15,8],[17,14,18,13],[10,3,11,4],[2,11,3,12],[18,8,19,9],[9,12,10,13]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (7,16,-8,-1)(13,2,-14,-3)(11,4,-12,-5)(18,5,-19,-6)(15,8,-16,-9)(6,9,-7,-10)(10,17,-11,-18)(3,12,-4,-13)(19,14,-20,-15)(1,20,-2,-17)
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,-17,10,-7)(-2,13,-4,11,17)(-3,-13)(-5,18,-11)(-6,-10,-18)(-8,15,-20,1)(-9,6,-19,-15)(-12,3,-14,19,5)(-16,7,9)(2,20,14)(4,12)(8,16)
Multiloop annotated with half-edges
12^2_317 annotated with half-edges