$12^{2}_{56}$ - Minimal pinning sets
Pinning sets for 12^2_56 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_56 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, 6, 9, 12}
5
[2, 2, 2, 3, 3]
2.40
a (minimal)
• {1, 2, 5, 6, 9, 12}
6
[2, 2, 2, 3, 3, 3]
2.50
b (minimal)
• {1, 3, 6, 8, 9, 11}
6
[2, 2, 2, 3, 3, 3]
2.50
c (minimal)
• {1, 2, 5, 6, 8, 9, 11}
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],[0,6,7,0],[0,4,4,1],[1,3,3,5],[1,4,8,6],[2,5,9,7],[2,6,9,8],[5,7,9,9],[6,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[10,20,1,11],[11,14,12,15],[19,9,20,10],[1,13,2,14],[12,2,13,3],[15,3,16,4],[4,18,5,19],[5,8,6,9],[16,6,17,7],[7,17,8,18]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (11,10,-12,-1)(12,3,-13,-4)(2,5,-3,-6)(15,6,-16,-7)(18,9,-19,-10)(8,19,-9,-20)(17,20,-18,-11)(4,13,-5,-14)(1,14,-2,-15)(7,16,-8,-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,-15,-7,-17,-11)(-2,-6,15)(-3,12,10,-19,8,16,6)(-4,-14,1,-12)(-5,2,14)(-8,-20,17)(-9,18,20)(-10,11,-18)(-13,4)(-16,7)(3,5,13)(9,19)
Multiloop annotated with half-edges
12^2_56 annotated with half-edges