$12^{2}_{296}$ - Minimal pinning sets
Pinning sets for 12^2_296 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_296 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 416
of which optimal: 1
of which minimal: 6
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.10977
on average over minimal pinning sets: 2.67222
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, 2, 3, 7}
4
[2, 2, 3, 3]
2.50
a (minimal)
• {1, 2, 4, 7, 11}
5
[2, 2, 3, 3, 4]
2.80
b (minimal)
• {1, 2, 7, 8, 11}
5
[2, 2, 3, 3, 3]
2.60
c (minimal)
• {1, 2, 5, 7, 11}
5
[2, 2, 3, 3, 4]
2.80
d (minimal)
• {1, 2, 6, 8, 9, 11}
6
[2, 2, 3, 3, 3, 3]
2.67
e (minimal)
• {1, 2, 3, 6, 8, 9}
6
[2, 2, 3, 3, 3, 3]
2.67
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
2.5
5
0
3
8
2.75
6
0
2
43
2.91
7
0
0
96
3.04
8
0
0
120
3.13
9
0
0
91
3.21
10
0
0
41
3.27
11
0
0
10
3.31
12
0
0
1
3.33
Total
1
5
410
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],[0,4,5,6],[0,6,6,7],[0,7,8,8],[0,9,5,1],[1,4,9,6],[1,5,2,2],[2,9,8,3],[3,7,9,3],[4,8,7,5]]
PD code (use to draw this multiloop with SnapPy): [[14,20,1,15],[15,11,16,12],[4,13,5,14],[19,7,20,8],[1,10,2,11],[16,2,17,3],[12,3,13,4],[5,9,6,8],[6,18,7,19],[9,17,10,18]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (9,14,-10,-1)(15,4,-16,-5)(5,2,-6,-3)(12,7,-13,-8)(17,8,-18,-9)(1,10,-2,-11)(18,13,-19,-14)(3,20,-4,-15)(11,16,-12,-17)(6,19,-7,-20)
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,-11,-17,-9)(-2,5,-16,11)(-3,-15,-5)(-4,15)(-6,-20,3)(-7,12,16,4,20)(-8,17,-12)(-10,1)(-13,18,8)(-14,9,-18)(-19,6,2,10,14)(7,19,13)
Multiloop annotated with half-edges
12^2_296 annotated with half-edges