$12^{2}_{27}$ - Minimal pinning sets
Pinning sets for 12^2_27
Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_27
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, 3, 5, 6, 9, 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, 3, 4, 4, 5, 6, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,5,0],[0,6,6,7],[0,8,4,4],[1,3,3,8],[1,7,6,6],[2,5,5,2],[2,5,9,9],[3,9,9,4],[7,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[4,20,1,5],[5,3,6,4],[8,19,9,20],[1,14,2,13],[2,12,3,13],[6,17,7,18],[18,7,19,8],[9,17,10,16],[14,11,15,12],[10,15,11,16]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (14,1,-15,-2)(11,16,-12,-17)(7,20,-8,-5)(4,5,-1,-6)(6,3,-7,-4)(19,8,-20,-9)(9,18,-10,-19)(15,10,-16,-11)(17,12,-18,-13)(2,13,-3,-14)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,14,-3,6)(-2,-14)(-4,-6)(-5,4,-7)(-8,19,-10,15,1,5)(-9,-19)(-11,-17,-13,2,-15)(-12,17)(-16,11)(-18,9,-20,7,3,13)(8,20)(10,18,12,16)
Multiloop annotated with half-edges
12^2_27 annotated with half-edges