$12^{2}_{37}$ - Minimal pinning sets
Pinning sets for 12^2_37 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_37 Pinning data
Pinning number of this multiloop: 6
Total number of pinning sets: 64
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: 2.85421
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, 4, 7, 11}
6
[2, 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
6
1
0
0
2.0
7
0
0
6
2.38
8
0
0
15
2.67
9
0
0
20
2.89
10
0
0
15
3.07
11
0
0
6
3.21
12
0
0
1
3.33
Total
1
0
63
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 2, 2, 4, 4, 4, 4, 6, 6]
Minimal region degree: 2
Is multisimple: Yes
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,4,0],[0,4,5,5],[0,6,6,4],[1,3,2,1],[2,7,8,2],[3,8,7,3],[5,6,9,9],[5,9,9,6],[7,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[10,20,1,11],[11,9,12,10],[19,7,20,8],[1,14,2,13],[8,12,9,13],[6,18,7,19],[14,3,15,2],[15,5,16,6],[17,3,18,4],[4,16,5,17]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (11,10,-12,-1)(19,2,-20,-3)(13,4,-14,-5)(15,6,-16,-7)(9,18,-10,-19)(5,16,-6,-17)(17,8,-18,-9)(1,20,-2,-11)(3,12,-4,-13)(7,14,-8,-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,-11)(-2,19,-10,11)(-3,-13,-5,-17,-9,-19)(-4,13)(-6,15,-8,17)(-7,-15)(-12,3,-20,1)(-14,7,-16,5)(-18,9)(2,20)(4,12,10,18,8,14)(6,16)
Multiloop annotated with half-edges
12^2_37 annotated with half-edges