$12^{3}_{43}$ - Minimal pinning sets
Pinning sets for 12^3_43 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^3_43 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 376
of which optimal: 1
of which minimal: 5
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.04011
on average over minimal pinning sets: 2.49
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, 3, 7, 11}
4
[2, 2, 2, 3]
2.25
a (minimal)
• {1, 3, 4, 6, 7}
5
[2, 2, 2, 3, 4]
2.60
b (minimal)
• {1, 3, 4, 7, 8}
5
[2, 2, 2, 3, 3]
2.40
c (minimal)
• {1, 2, 3, 4, 7}
5
[2, 2, 2, 3, 3]
2.40
d (minimal)
• {1, 3, 4, 7, 9}
5
[2, 2, 2, 3, 5]
2.80
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
4
8
2.57
6
0
0
46
2.8
7
0
0
90
2.97
8
0
0
105
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
4
371
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: [[1,2,3,4],[0,5,5,2],[0,1,5,6],[0,7,8,8],[0,8,9,5],[1,4,2,1],[2,9,9,7],[3,6,9,8],[3,7,4,3],[4,7,6,6]]
PD code (use to draw this multiloop with SnapPy): [[8,14,1,9],[9,6,10,5],[7,4,8,5],[13,20,14,15],[1,12,2,11],[6,11,7,10],[17,3,18,4],[15,18,16,19],[19,12,20,13],[2,16,3,17]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (10,1,-11,-2)(5,2,-6,-3)(3,20,-4,-15)(15,4,-16,-5)(16,7,-17,-8)(12,17,-13,-18)(6,19,-7,-20)(14,11,-9,-12)(18,13,-19,-14)(8,9,-1,-10)
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)(-2,5,-16,-8,-10)(-3,-15,-5)(-4,15)(-6,-20,3)(-7,16,4,20)(-9,8,-17,12)(-11,14,-19,6,2)(-12,-18,-14)(-13,18)(1,9,11)(7,19,13,17)
Multiloop annotated with half-edges
12^3_43 annotated with half-edges