- Minimal pinning sets
Pinning sets for 12^3_27 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^3_27 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, 3, 5, 7, 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, 3, 3, 4, 7, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,3,3,4],[0,5,6,0],[0,4,1,1],[1,3,7,7],[2,8,6,6],[2,5,5,9],[4,9,8,4],[5,7,9,9],[6,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[3,8,4,1],[2,16,3,9],[7,4,8,5],[1,10,2,9],[10,15,11,16],[5,17,6,20],[6,19,7,20],[14,11,15,12],[17,14,18,13],[18,12,19,13]]
Permutation representation (action on half-edges):
Vertex permutation (5,2,-6,-3)(1,6,-2,-7)(15,12,-16,-13)(20,13,-17,-14)(14,19,-15,-20)(11,16,-12,-9)(8,9,-1,-10)(10,7,-11,-8)(4,17,-5,-18)(18,3,-19,-4)
Edge permutation (-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 (-1,-7,10)(-2,5,17,13,-16,11,7)(-3,18,-5)(-4,-18)(-6,1,9,-12,15,19,3)(-8,-10)(-9,8,-11)(-13,20,-15)(-14,-20)(-17,4,-19,14)(2,6)(12,16)
Multiloop annotated with half-edges
12^3_27 annotated with half-edges