$12^{3}_{25}$ - Minimal pinning sets
Pinning sets for 12^3_25
Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^3_25
Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 256
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: 2.96564
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)
•
{2, 6, 7, 11}
4
[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
4
1
0
0
2.0
5
0
0
8
2.4
6
0
0
28
2.67
7
0
0
56
2.86
8
0
0
70
3.0
9
0
0
56
3.11
10
0
0
28
3.2
11
0
0
8
3.27
12
0
0
1
3.33
Total
1
0
255
Other information about this multiloop
Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 5, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,4,5],[0,6,6,3],[0,2,7,8],[0,5,1,1],[1,4,8,9],[2,7,7,2],[3,6,6,9],[3,9,9,5],[5,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[3,8,4,1],[2,14,3,9],[7,20,8,15],[4,20,5,19],[1,10,2,9],[10,13,11,14],[15,6,16,7],[5,16,6,17],[12,18,13,19],[11,18,12,17]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (17,4,-18,-5)(1,6,-2,-7)(11,14,-12,-9)(8,9,-1,-10)(10,7,-11,-8)(19,12,-20,-13)(13,20,-14,-15)(15,2,-16,-3)(5,16,-6,-17)(3,18,-4,-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,-7,10)(-2,15,-14,11,7)(-3,-19,-13,-15)(-4,17,-6,1,9,-12,19)(-5,-17)(-8,-10)(-9,8,-11)(-16,5,-18,3)(-20,13)(2,6,16)(4,18)(12,14,20)
Multiloop annotated with half-edges
12^3_25 annotated with half-edges