$12^{2}_{213}$ - Minimal pinning sets
Pinning sets for 12^2_213 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_213 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 192
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: 2.96906
on average over minimal pinning sets: 2.2
on average over optimal pinning sets: 2.2
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 4, 7, 11}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 2, 5, 7, 11}
5
[2, 2, 2, 2, 3]
2.20
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
2
0
0
2.2
6
0
0
13
2.54
7
0
0
36
2.78
8
0
0
55
2.95
9
0
0
50
3.09
10
0
0
27
3.19
11
0
0
8
3.27
12
0
0
1
3.33
Total
2
0
190
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,2],[0,1,6,6],[0,7,8,4],[0,3,8,5],[1,4,6,1],[2,5,7,2],[3,6,9,9],[3,9,9,4],[7,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[9,16,10,1],[8,20,9,17],[15,19,16,20],[10,5,11,6],[1,6,2,7],[17,7,18,8],[18,14,19,15],[4,13,5,14],[11,3,12,2],[12,3,13,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (16,9,-1,-10)(10,1,-11,-2)(8,3,-9,-4)(15,4,-16,-5)(18,5,-19,-6)(2,11,-3,-12)(19,12,-20,-13)(7,14,-8,-15)(13,20,-14,-17)(6,17,-7,-18)
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,-12,19,5,-16,-10)(-3,8,14,20,12)(-4,15,-8)(-5,18,-7,-15)(-6,-18)(-9,16,4)(-11,2)(-13,-17,6,-19)(-14,7,17)(-20,13)(1,9,3,11)
Multiloop annotated with half-edges
12^2_213 annotated with half-edges