$12^{3}_{15}$ - Minimal pinning sets
Pinning sets for 12^3_15
Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^3_15
Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 192
of which optimal: 2
of which minimal: 2
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.96906
on average over minimal pinning sets: 2.2
on average over optimal pinning sets: 2.2
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, 3]
2.20
B (optimal)
•
{1, 2, 3, 7, 11}
5
[2, 2, 2, 2, 3]
2.20
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
2
0
0
2.2
6
0
0
13
2.54
7
0
0
36
2.78
8
0
0
55
2.95
9
0
0
50
3.09
10
0
0
27
3.19
11
0
0
8
3.27
12
0
0
1
3.33
Total
2
0
190
Other information about this multiloop
Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,5,0],[0,5,5,6],[0,6,4,4],[1,3,3,7],[1,8,2,2],[2,9,7,3],[4,6,9,8],[5,7,9,9],[6,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[4,8,1,5],[5,3,6,4],[7,20,8,9],[1,13,2,14],[14,2,15,3],[6,10,7,9],[12,19,13,20],[15,19,16,18],[10,18,11,17],[11,16,12,17]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (5,4,-6,-1)(18,1,-19,-2)(2,17,-3,-18)(3,16,-4,-5)(14,7,-15,-8)(8,13,-9,-14)(9,6,-10,-7)(15,10,-16,-11)(20,11,-17,-12)(12,19,-13,-20)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,18,-3,-5)(-2,-18)(-4,5)(-6,9,13,19,1)(-7,14,-9)(-8,-14)(-10,15,7)(-11,20,-13,8,-15)(-12,-20)(-16,3,17,11)(-17,2,-19,12)(4,16,10,6)
Multiloop annotated with half-edges
12^3_15 annotated with half-edges