$12^{2}_{34}$ - Minimal pinning sets
Pinning sets for 12^2_34 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_34 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 256
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: 2.96564
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, 7}
4
[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
4
1
0
0
2.0
5
0
0
8
2.4
6
0
0
28
2.67
7
0
0
56
2.86
8
0
0
70
3.0
9
0
0
56
3.11
10
0
0
28
3.2
11
0
0
8
3.27
12
0
0
1
3.33
Total
1
0
255
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,5,0],[0,5,3,3],[0,2,2,4],[1,3,6,7],[1,8,6,2],[4,5,9,9],[4,9,8,8],[5,7,7,9],[6,8,7,6]]
PD code (use to draw this multiloop with SnapPy): [[14,3,1,4],[4,13,5,14],[2,20,3,15],[1,20,2,19],[12,18,13,19],[5,16,6,15],[6,11,7,12],[17,9,18,10],[16,9,17,8],[10,7,11,8]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (12,1,-13,-2)(8,5,-9,-6)(6,17,-7,-18)(18,7,-19,-8)(4,9,-5,-10)(2,11,-3,-12)(10,19,-11,-20)(20,13,-15,-14)(14,15,-1,-16)(16,3,-17,-4)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,12,-3,16)(-2,-12)(-4,-10,-20,-14,-16)(-5,8,-19,10)(-6,-18,-8)(-7,18)(-9,4,-17,6)(-11,2,-13,20)(-15,14)(1,15,13)(3,11,19,7,17)(5,9)
Multiloop annotated with half-edges
12^2_34 annotated with half-edges