$12^{2}_{59}$ - Minimal pinning sets
Pinning sets for 12^2_59 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_59 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 200
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: 3.03442
on average over minimal pinning sets: 2.49286
on average over optimal pinning sets: 2.4
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 4, 9, 11}
5
[2, 2, 2, 3, 3]
2.40
a (minimal)
• {1, 2, 3, 5, 9, 11}
6
[2, 2, 2, 3, 3, 3]
2.50
b (minimal)
• {1, 3, 4, 7, 11, 12}
6
[2, 2, 2, 3, 3, 3]
2.50
c (minimal)
• {1, 2, 3, 5, 7, 11, 12}
7
[2, 2, 2, 3, 3, 3, 3]
2.57
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
1
0
0
2.4
6
0
2
7
2.63
7
0
1
31
2.82
8
0
0
58
2.99
9
0
0
58
3.12
10
0
0
32
3.23
11
0
0
9
3.29
12
0
0
1
3.33
Total
1
3
196
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 5, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,3,4,5],[0,6,3,0],[0,2,6,1],[1,7,8,5],[1,4,8,9],[2,7,7,3],[4,6,6,9],[4,9,9,5],[5,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[12,20,1,13],[13,10,14,9],[19,11,20,12],[1,11,2,10],[14,4,15,5],[5,8,6,9],[18,2,19,3],[3,17,4,18],[15,7,16,8],[6,16,7,17]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (18,1,-19,-2)(19,4,-20,-5)(2,5,-3,-6)(6,17,-7,-18)(7,10,-8,-11)(15,8,-16,-9)(14,11,-15,-12)(3,20,-4,-13)(12,13,-1,-14)(9,16,-10,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,18,-7,-11,14)(-2,-6,-18)(-3,-13,12,-15,-9,-17,6)(-4,19,1,13)(-5,2,-19)(-8,15,11)(-10,7,17)(-12,-14)(-16,9)(-20,3,5)(4,20)(8,10,16)
Multiloop annotated with half-edges
12^2_59 annotated with half-edges