$12^{2}_{164}$ - Minimal pinning sets
Pinning sets for 12^2_164 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_164 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, 5, 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, 4, 5, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,2],[0,1,6,6],[0,7,7,4],[0,3,8,5],[1,4,6,1],[2,5,8,2],[3,9,9,3],[4,9,9,6],[7,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[9,16,10,1],[8,20,9,17],[15,19,16,20],[10,5,11,6],[1,6,2,7],[17,7,18,8],[18,14,19,15],[4,11,5,12],[2,13,3,14],[12,3,13,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (9,16,-10,-1)(7,2,-8,-3)(14,3,-15,-4)(17,4,-18,-5)(1,8,-2,-9)(15,10,-16,-11)(18,11,-19,-12)(6,13,-7,-14)(12,19,-13,-20)(5,20,-6,-17)
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,-9)(-2,7,13,19,11,-16,9)(-3,14,-7)(-4,17,-6,-14)(-5,-17)(-8,1,-10,15,3)(-11,18,4,-15)(-12,-20,5,-18)(-13,6,20)(-19,12)(2,8)(10,16)
Multiloop annotated with half-edges
12^2_164 annotated with half-edges