$12^{3}_{6}$ - Minimal pinning sets
Pinning sets for 12^3_6 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^3_6 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, 4, 6, 9}
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: [[1,2,3,4],[0,4,4,5],[0,6,6,7],[0,7,8,8],[0,5,1,1],[1,4,8,6],[2,5,9,2],[2,9,9,3],[3,9,5,3],[6,8,7,7]]
PD code (use to draw this multiloop with SnapPy): [[3,12,4,1],[2,7,3,8],[11,16,12,13],[4,17,5,20],[1,9,2,8],[9,6,10,7],[13,10,14,11],[15,17,16,18],[5,19,6,20],[14,19,15,18]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (3,12,-4,-1)(8,1,-9,-2)(2,7,-3,-8)(13,4,-14,-5)(9,6,-10,-7)(5,16,-6,-13)(19,14,-20,-15)(11,20,-12,-17)(17,10,-18,-11)(15,18,-16,-19)
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,-3)(-2,-8)(-4,13,-6,9,1)(-5,-13)(-7,2,-9)(-10,17,-12,3,7)(-11,-17)(-14,19,-16,5)(-15,-19)(-18,15,-20,11)(4,12,20,14)(6,16,18,10)
Multiloop annotated with half-edges
12^3_6 annotated with half-edges