$12^{3}_{26}$ - Minimal pinning sets
Pinning sets for 12^3_26
Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^3_26
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, 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, 3, 3, 4, 7, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,3,3,4],[0,5,5,0],[0,4,1,1],[1,3,6,7],[2,8,9,2],[4,9,7,7],[4,6,6,8],[5,7,9,9],[5,8,8,6]]
PD code (use to draw this multiloop with SnapPy): [[3,10,4,1],[2,16,3,11],[9,4,10,5],[1,12,2,11],[12,15,13,16],[5,8,6,9],[14,20,15,17],[13,20,14,19],[7,18,8,19],[6,18,7,17]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (7,2,-8,-3)(3,6,-4,-7)(1,8,-2,-9)(13,16,-14,-11)(10,11,-1,-12)(12,9,-13,-10)(5,18,-6,-19)(19,14,-20,-15)(15,20,-16,-17)(17,4,-18,-5)
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,-9,12)(-2,7,-4,17,-16,13,9)(-3,-7)(-5,-19,-15,-17)(-6,3,-8,1,11,-14,19)(-10,-12)(-11,10,-13)(-18,5)(-20,15)(2,8)(4,6,18)(14,16,20)
Multiloop annotated with half-edges
12^3_26 annotated with half-edges