$12^{2}_{98}$ - Minimal pinning sets
Pinning sets for 12^2_98 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_98 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, 4, 7, 8, 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, 3, 3, 5, 6, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,4,5],[0,6,6,3],[0,2,7,7],[0,5,1,1],[1,4,7,8],[2,9,9,2],[3,8,5,3],[5,7,9,9],[6,8,8,6]]
PD code (use to draw this multiloop with SnapPy): [[3,14,4,1],[2,7,3,8],[13,20,14,15],[4,20,5,19],[1,9,2,8],[9,6,10,7],[15,12,16,13],[5,18,6,19],[10,18,11,17],[11,16,12,17]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (7,4,-8,-5)(14,5,-1,-6)(6,13,-7,-14)(17,10,-18,-11)(1,12,-2,-13)(19,8,-20,-9)(3,20,-4,-15)(15,2,-16,-3)(11,16,-12,-17)(9,18,-10,-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,-13,6)(-2,15,-4,7,13)(-3,-15)(-5,14,-7)(-6,-14)(-8,19,-10,17,-12,1,5)(-9,-19)(-11,-17)(-16,11,-18,9,-20,3)(2,12,16)(4,20,8)(10,18)
Multiloop annotated with half-edges
12^2_98 annotated with half-edges