$12^{3}_{23}$ - Minimal pinning sets
Pinning sets for 12^3_23 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^3_23 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, 3, 5, 7}
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, 5, 5, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,4,5],[0,6,7,7],[0,8,8,9],[0,5,1,1],[1,4,9,6],[2,5,9,9],[2,8,8,2],[3,7,7,3],[3,6,6,5]]
PD code (use to draw this multiloop with SnapPy): [[3,8,4,1],[2,14,3,9],[7,20,8,15],[4,18,5,17],[1,10,2,9],[10,13,11,14],[15,11,16,12],[19,6,20,7],[18,6,19,5],[12,16,13,17]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (8,9,-1,-10)(10,1,-11,-2)(6,3,-7,-4)(12,17,-13,-18)(16,19,-17,-20)(5,20,-6,-15)(15,4,-16,-5)(14,7,-9,-8)(2,11,-3,-12)(18,13,-19,-14)
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)(-2,-12,-18,-14,-8,-10)(-3,6,20,-17,12)(-4,15,-6)(-5,-15)(-7,14,-19,16,4)(-9,8)(-11,2)(-13,18)(-16,-20,5)(1,9,7,3,11)(13,17,19)
Multiloop annotated with half-edges
12^3_23 annotated with half-edges