$12^{2}_{31}$ - Minimal pinning sets
Pinning sets for 12^2_31 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_31 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, 4, 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, 4, 4, 4, 4, 4, 4, 6]
Minimal region degree: 2
Is multisimple: Yes
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,4,4,5],[0,5,5,0],[0,6,7,4],[1,3,8,1],[1,8,2,2],[3,8,9,9],[3,9,9,8],[4,7,6,5],[6,7,7,6]]
PD code (use to draw this multiloop with SnapPy): [[10,20,1,11],[11,7,12,8],[19,9,20,10],[1,14,2,13],[6,12,7,13],[8,18,9,19],[14,5,15,4],[2,16,3,17],[17,5,18,6],[15,3,16,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (11,10,-12,-1)(15,2,-16,-3)(13,4,-14,-5)(19,8,-20,-9)(9,18,-10,-19)(1,16,-2,-17)(17,6,-18,-7)(7,20,-8,-11)(3,12,-4,-13)(5,14,-6,-15)
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,-17,-7,-11)(-2,15,-6,17)(-3,-13,-5,-15)(-4,13)(-8,19,-10,11)(-9,-19)(-12,3,-16,1)(-14,5)(-18,9,-20,7)(2,16)(4,12,10,18,6,14)(8,20)
Multiloop annotated with half-edges
12^2_31 annotated with half-edges