$12^{3}_{63}$ - Minimal pinning sets
Pinning sets for 12^3_63 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^3_63 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, 5, 5, 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],[1,9,2,2],[2,9,7,7],[3,6,6,3],[4,9,9,4],[5,8,8,6]]
PD code (use to draw this multiloop with SnapPy): [[8,12,1,9],[9,7,10,8],[11,20,12,13],[1,18,2,17],[6,16,7,17],[10,14,11,13],[19,3,20,4],[18,3,19,2],[15,5,16,6],[14,5,15,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (8,9,-1,-10)(10,1,-11,-2)(2,13,-3,-14)(14,3,-15,-4)(18,5,-19,-6)(6,17,-7,-18)(4,15,-5,-16)(16,7,-9,-8)(20,11,-17,-12)(12,19,-13,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,10)(-2,-14,-4,-16,-8,-10)(-3,14)(-5,18,-7,16)(-6,-18)(-9,8)(-11,20,-13,2)(-12,-20)(-15,4)(-17,6,-19,12)(1,9,7,17,11)(3,13,19,5,15)
Multiloop annotated with half-edges
12^3_63 annotated with half-edges