$12^{3}_{60}$ - Minimal pinning sets
Pinning sets for 12^3_60 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^3_60 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, 5, 8, 9, 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, 3, 4, 4, 5, 6, 6]
Minimal region degree: 2
Is multisimple: Yes
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,5,0],[0,6,6,3],[0,2,7,4],[1,3,8,8],[1,7,9,9],[2,9,9,2],[3,5,8,8],[4,7,7,4],[5,6,6,5]]
PD code (use to draw this multiloop with SnapPy): [[6,12,1,7],[7,5,8,6],[11,20,12,13],[1,20,2,19],[4,18,5,19],[8,16,9,15],[13,10,14,11],[2,16,3,17],[17,3,18,4],[9,14,10,15]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (8,1,-9,-2)(16,3,-17,-4)(12,19,-13,-20)(4,15,-5,-16)(14,17,-7,-18)(6,7,-1,-8)(2,9,-3,-10)(10,5,-11,-6)(20,11,-15,-12)(18,13,-19,-14)
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)(-2,-10,-6,-8)(-3,16,-5,10)(-4,-16)(-7,6,-11,20,-13,18)(-9,2)(-12,-20)(-14,-18)(-15,4,-17,14,-19,12)(1,7,17,3,9)(5,15,11)(13,19)
Multiloop annotated with half-edges
12^3_60 annotated with half-edges