$12^{3}_{40}$ - Minimal pinning sets
Pinning sets for 12^3_40
Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^3_40
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, 3, 4, 6, 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, 4, 4, 4, 5, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,4,4,5],[0,5,6,6],[0,7,8,0],[1,8,9,1],[1,9,9,2],[2,9,7,2],[3,6,8,8],[3,7,7,4],[4,6,5,5]]
PD code (use to draw this multiloop with SnapPy): [[12,5,1,6],[6,13,7,16],[11,20,12,17],[4,1,5,2],[13,8,14,7],[15,17,16,18],[19,10,20,11],[2,10,3,9],[3,8,4,9],[14,19,15,18]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (9,2,-10,-3)(4,13,-5,-14)(12,5,-1,-6)(6,11,-7,-12)(20,7,-17,-8)(1,10,-2,-11)(16,3,-13,-4)(19,14,-20,-15)(8,17,-9,-18)(15,18,-16,-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,-11,6)(-2,9,17,7,11)(-3,16,18,-9)(-4,-14,19,-16)(-5,12,-7,20,14)(-6,-12)(-8,-18,15,-20)(-10,1,5,13,3)(-13,4)(-15,-19)(-17,8)(2,10)
Multiloop annotated with half-edges
12^3_40 annotated with half-edges