$12^{2}_{157}$ - Minimal pinning sets
Pinning sets for 12^2_157 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_157 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)
• {2, 3, 6, 9, 11}
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, 4, 4, 4, 5, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,6],[0,6,7,7],[0,7,4,4],[0,3,3,5],[1,4,8,1],[1,9,9,2],[2,8,3,2],[5,7,9,9],[6,8,8,6]]
PD code (use to draw this multiloop with SnapPy): [[10,3,1,4],[4,11,5,20],[15,9,16,10],[2,7,3,8],[1,7,2,6],[11,6,12,5],[14,19,15,20],[8,16,9,17],[12,17,13,18],[18,13,19,14]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (8,1,-9,-2)(3,12,-4,-13)(4,9,-5,-10)(10,5,-1,-6)(16,7,-17,-8)(19,14,-20,-15)(6,17,-7,-18)(13,18,-14,-19)(15,20,-16,-11)(11,2,-12,-3)
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,-17,6)(-2,11,-16,-8)(-3,-13,-19,-15,-11)(-4,-10,-6,-18,13)(-5,10)(-7,16,20,14,18)(-9,4,12,2)(-12,3)(-14,19)(-20,15)(1,5,9)(7,17)
Multiloop annotated with half-edges
12^2_157 annotated with half-edges