$12^{3}_{7}$ - Minimal pinning sets
Pinning sets for 12^3_7 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^3_7 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, 3, 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, 5, 5, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,5,0],[0,6,6,7],[0,8,4,4],[1,3,3,8],[1,7,9,9],[2,9,9,2],[2,5,8,8],[3,7,7,4],[5,6,6,5]]
PD code (use to draw this multiloop with SnapPy): [[4,10,1,5],[5,3,6,4],[9,20,10,11],[1,18,2,17],[2,16,3,17],[6,14,7,13],[11,8,12,9],[14,19,15,20],[18,15,19,16],[7,12,8,13]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (16,1,-17,-2)(10,19,-11,-20)(2,15,-3,-16)(18,11,-19,-12)(12,17,-13,-18)(7,14,-8,-5)(4,5,-1,-6)(6,3,-7,-4)(13,8,-14,-9)(20,9,-15,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,16,-3,6)(-2,-16)(-4,-6)(-5,4,-7)(-8,13,17,1,5)(-9,20,-11,18,-13)(-10,-20)(-12,-18)(-14,7,3,15,9)(-15,2,-17,12,-19,10)(8,14)(11,19)
Multiloop annotated with half-edges
12^3_7 annotated with half-edges