$12^{3}_{39}$ - Minimal pinning sets
Pinning sets for 12^3_39
Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^3_39
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, 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, 4, 4, 4, 5, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,3,4,4],[0,5,6,6],[0,6,1,0],[1,7,8,1],[2,9,9,7],[2,7,3,2],[4,6,5,8],[4,7,9,9],[5,8,8,5]]
PD code (use to draw this multiloop with SnapPy): [[4,14,1,5],[5,13,6,12],[3,20,4,15],[13,1,14,2],[6,11,7,12],[15,9,16,10],[19,2,20,3],[10,18,11,19],[7,18,8,17],[8,16,9,17]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (11,2,-12,-3)(10,3,-5,-4)(6,17,-7,-18)(14,19,-15,-20)(20,13,-11,-14)(1,12,-2,-13)(8,15,-9,-16)(18,9,-19,-10)(4,5,-1,-6)(16,7,-17,-8)
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,-13,20,-15,8,-17,6)(-2,11,13)(-3,10,-19,14,-11)(-4,-6,-18,-10)(-5,4)(-7,16,-9,18)(-8,-16)(-12,1,5,3)(-14,-20)(2,12)(7,17)(9,15,19)
Multiloop annotated with half-edges
12^3_39 annotated with half-edges