$12^{3}_{36}$ - Minimal pinning sets
Pinning sets for 12^3_36
Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^3_36
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)
•
{1, 3, 5, 7}
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, 5, 5, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,3,4,5],[0,6,6,7],[0,8,1,0],[1,9,5,5],[1,4,4,6],[2,5,7,2],[2,6,9,8],[3,7,9,9],[4,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[8,14,1,9],[9,13,10,12],[7,20,8,15],[13,1,14,2],[10,4,11,5],[5,11,6,12],[15,6,16,7],[16,19,17,20],[2,17,3,18],[18,3,19,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (9,8,-10,-1)(15,2,-16,-3)(12,5,-13,-6)(18,7,-19,-8)(1,10,-2,-11)(11,14,-12,-9)(4,13,-5,-14)(6,17,-7,-18)(16,19,-17,-20)(3,20,-4,-15)
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,-9)(-2,15,-4,-14,11)(-3,-15)(-5,12,14)(-6,-18,-8,9,-12)(-7,18)(-10,1)(-13,4,20,-17,6)(-16,-20,3)(-19,16,2,10,8)(5,13)(7,17,19)
Multiloop annotated with half-edges
12^3_36 annotated with half-edges