$12^{3}_{31}$ - Minimal pinning sets
Pinning sets for 12^3_31
Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^3_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, 3, 7, 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, 3, 4, 5, 5, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,2],[0,3,4,0],[0,5,5,0],[1,5,6,7],[1,8,9,5],[2,4,3,2],[3,9,7,7],[3,6,6,8],[4,7,9,9],[4,8,8,6]]
PD code (use to draw this multiloop with SnapPy): [[8,16,1,9],[9,7,10,8],[15,1,16,2],[3,6,4,7],[10,13,11,14],[2,14,3,15],[5,20,6,17],[4,20,5,19],[12,18,13,19],[11,18,12,17]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (1,4,-2,-5)(12,5,-13,-6)(10,7,-11,-8)(13,16,-14,-9)(8,9,-1,-10)(6,11,-7,-12)(17,2,-18,-3)(3,18,-4,-19)(19,14,-20,-15)(15,20,-16,-17)
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,-5,12,-7,10)(-2,17,-16,13,5)(-3,-19,-15,-17)(-4,1,9,-14,19)(-6,-12)(-8,-10)(-9,8,-11,6,-13)(-18,3)(-20,15)(2,4,18)(7,11)(14,16,20)
Multiloop annotated with half-edges
12^3_31 annotated with half-edges