$12^{2}_{7}$ - Minimal pinning sets
Pinning sets for 12^2_7 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_7 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 128
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: 2.90623
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, 6, 9}
5
[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
5
1
0
0
2.0
6
0
0
7
2.38
7
0
0
21
2.65
8
0
0
35
2.86
9
0
0
35
3.02
10
0
0
21
3.14
11
0
0
7
3.25
12
0
0
1
3.33
Total
1
0
127
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 2, 3, 3, 4, 4, 5, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,4,4,5],[0,6,6,7],[0,8,4,0],[1,3,5,1],[1,4,9,9],[2,9,7,2],[2,6,8,8],[3,7,7,9],[5,8,6,5]]
PD code (use to draw this multiloop with SnapPy): [[16,20,1,17],[17,11,18,12],[15,4,16,5],[19,1,20,2],[10,18,11,19],[12,10,13,9],[5,14,6,15],[6,3,7,4],[2,7,3,8],[13,8,14,9]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (11,16,-12,-1)(14,3,-15,-4)(9,4,-10,-5)(5,8,-6,-9)(20,7,-17,-8)(1,10,-2,-11)(18,13,-19,-14)(2,15,-3,-16)(6,17,-7,-18)(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,-11)(-2,-16,11)(-3,14,-19,12,16)(-4,9,-6,-18,-14)(-5,-9)(-7,20,-13,18)(-8,5,-10,1,-12,-20)(-15,2,10,4)(-17,6,8)(3,15)(7,17)(13,19)
Multiloop annotated with half-edges
12^2_7 annotated with half-edges