$12^{2}_{135}$ - Minimal pinning sets
Pinning sets for 12^2_135 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_135 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 128
of which optimal: 1
of which minimal: 1
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.90623
on average over minimal pinning sets: 2.0
on average over optimal pinning sets: 2.0
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 4, 5, 9}
5
[2, 2, 2, 2, 2]
2.00
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
1
0
0
2.0
6
0
0
7
2.38
7
0
0
21
2.65
8
0
0
35
2.86
9
0
0
35
3.02
10
0
0
21
3.14
11
0
0
7
3.25
12
0
0
1
3.33
Total
1
0
127
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 2, 3, 3, 4, 4, 5, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,3,4,5],[0,6,6,7],[0,7,1,0],[1,8,5,5],[1,4,4,6],[2,5,8,2],[2,9,9,3],[4,9,9,6],[7,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[14,20,1,15],[15,19,16,18],[6,13,7,14],[19,1,20,2],[16,10,17,11],[11,17,12,18],[12,5,13,6],[7,3,8,2],[4,9,5,10],[3,9,4,8]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (8,1,-9,-2)(4,13,-5,-14)(14,5,-1,-6)(15,6,-16,-7)(2,9,-3,-10)(18,11,-19,-12)(12,3,-13,-4)(7,16,-8,-17)(17,20,-18,-15)(10,19,-11,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,8,16,6)(-2,-10,-20,17,-8)(-3,12,-19,10)(-4,-14,-6,15,-18,-12)(-5,14)(-7,-17,-15)(-9,2)(-11,18,20)(-13,4)(-16,7)(1,5,13,3,9)(11,19)
Multiloop annotated with half-edges
12^2_135 annotated with half-edges