$12^{3}_{49}$ - Minimal pinning sets
Pinning sets for 12^3_49 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^3_49 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, 2, 5, 7}
4
[2, 2, 2, 3]
2.25
a (minimal)
• {1, 3, 4, 5, 7}
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, 4, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,6],[0,6,7,7],[0,8,9,4],[0,3,9,5],[1,4,6,1],[1,5,7,2],[2,6,8,2],[3,7,9,9],[3,8,8,4]]
PD code (use to draw this multiloop with SnapPy): [[6,14,1,7],[7,15,8,20],[17,5,18,6],[13,10,14,11],[1,10,2,9],[15,9,16,8],[16,19,17,20],[4,18,5,19],[11,4,12,3],[12,2,13,3]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (7,6,-8,-1)(1,14,-2,-7)(20,3,-15,-4)(11,18,-12,-19)(12,5,-13,-6)(8,13,-9,-14)(4,9,-5,-10)(2,15,-3,-16)(19,16,-20,-17)(17,10,-18,-11)
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,-7)(-2,-16,19,-12,-6,7)(-3,20,16)(-4,-10,17,-20)(-5,12,18,10)(-8,-14,1)(-9,4,-15,2,14)(-11,-19,-17)(-13,8,6)(-18,11)(3,15)(5,9,13)
Multiloop annotated with half-edges
12^3_49 annotated with half-edges