$12^{3}_{62}$ - Minimal pinning sets
Pinning sets for 12^3_62 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^3_62 Pinning data
Pinning number of this multiloop: 6
Total number of pinning sets: 64
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: 2.85421
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, 3, 4, 7, 11}
6
[2, 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
6
1
0
0
2.0
7
0
0
6
2.38
8
0
0
15
2.67
9
0
0
20
2.89
10
0
0
15
3.07
11
0
0
6
3.21
12
0
0
1
3.33
Total
1
0
63
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 2, 2, 4, 4, 4, 4, 6, 6]
Minimal region degree: 2
Is multisimple: Yes
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,5,0],[0,5,5,6],[0,7,7,4],[1,3,8,8],[1,8,2,2],[2,8,9,9],[3,9,9,3],[4,6,5,4],[6,7,7,6]]
PD code (use to draw this multiloop with SnapPy): [[10,14,1,11],[11,9,12,10],[13,5,14,6],[1,15,2,20],[8,19,9,20],[12,7,13,6],[4,17,5,18],[15,3,16,2],[18,7,19,8],[16,3,17,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (10,11,-1,-12)(12,1,-13,-2)(18,3,-19,-4)(4,17,-5,-18)(14,5,-15,-6)(20,7,-17,-8)(8,19,-9,-20)(2,13,-3,-14)(6,15,-7,-16)(16,9,-11,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,12)(-2,-14,-6,-16,-10,-12)(-3,18,-5,14)(-4,-18)(-7,20,-9,16)(-8,-20)(-11,10)(-13,2)(-15,6)(-17,4,-19,8)(1,11,9,19,3,13)(5,17,7,15)
Multiloop annotated with half-edges
12^3_62 annotated with half-edges