$12^{3}_{32}$ - Minimal pinning sets
Pinning sets for 12^3_32
Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^3_32
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, 5, 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: [[1,1,2,3],[0,4,4,0],[0,5,5,6],[0,7,8,4],[1,3,5,1],[2,4,9,2],[2,9,9,7],[3,6,8,8],[3,7,7,9],[5,8,6,6]]
PD code (use to draw this multiloop with SnapPy): [[7,12,8,1],[11,6,12,7],[8,13,9,16],[1,4,2,5],[5,10,6,11],[13,10,14,9],[15,20,16,17],[3,19,4,20],[2,19,3,18],[14,18,15,17]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (10,1,-11,-2)(8,3,-9,-4)(6,17,-7,-18)(2,7,-3,-8)(4,9,-5,-10)(20,5,-17,-6)(16,11,-13,-12)(12,13,-1,-14)(19,14,-20,-15)(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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,10,-5,20,14)(-2,-8,-4,-10)(-3,8)(-6,-18,15,-20)(-7,2,-11,16,18)(-9,4)(-12,-14,19,-16)(-13,12)(-15,-19)(-17,6)(1,13,11)(3,7,17,5,9)
Multiloop annotated with half-edges
12^3_32 annotated with half-edges