$12^{3}_{73}$ - Minimal pinning sets
Pinning sets for 12^3_73 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^3_73 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 448
of which optimal: 2
of which minimal: 2
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.10492
on average over minimal pinning sets: 2.5
on average over optimal pinning sets: 2.5
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 4, 5, 7}
4
[2, 2, 3, 3]
2.50
B (optimal)
• {2, 3, 5, 7}
4
[2, 2, 3, 3]
2.50
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
2
0
0
2.5
5
0
0
16
2.75
6
0
0
55
2.92
7
0
0
106
3.05
8
0
0
125
3.14
9
0
0
92
3.21
10
0
0
41
3.27
11
0
0
10
3.31
12
0
0
1
3.33
Total
2
0
446
Other information about this multiloop Properties
Region degree sequence: [2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,2],[0,1,5,6],[0,6,6,7],[0,8,5,1],[1,4,9,2],[2,7,3,3],[3,6,9,8],[4,7,9,9],[5,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[4,14,1,5],[5,10,6,11],[11,3,12,4],[13,20,14,15],[1,9,2,10],[6,2,7,3],[12,16,13,15],[16,19,17,20],[17,8,18,9],[7,18,8,19]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (5,4,-6,-1)(10,1,-11,-2)(20,7,-15,-8)(3,8,-4,-9)(14,9,-5,-10)(2,13,-3,-14)(6,15,-7,-16)(19,16,-20,-17)(12,17,-13,-18)(18,11,-19,-12)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,10,-5)(-2,-14,-10)(-3,-9,14)(-4,5,9)(-6,-16,19,11,1)(-7,20,16)(-8,3,13,17,-20)(-11,18,-13,2)(-12,-18)(-15,6,4,8)(-17,12,-19)(7,15)
Multiloop annotated with half-edges
12^3_73 annotated with half-edges