$12^{3}_{1}$ - Minimal pinning sets
Pinning sets for 12^3_1 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^3_1 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)
• {2, 4, 5, 9}
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, 4, 4, 5, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,4,5],[0,6,6,7],[0,7,8,8],[0,9,1,1],[1,9,9,6],[2,5,7,2],[2,6,8,3],[3,7,9,3],[4,8,5,5]]
PD code (use to draw this multiloop with SnapPy): [[3,10,4,1],[2,14,3,11],[9,20,10,15],[4,17,5,18],[1,12,2,11],[7,13,8,14],[15,8,16,9],[16,19,17,20],[5,19,6,18],[12,6,13,7]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (13,4,-14,-5)(19,6,-20,-7)(1,8,-2,-9)(5,14,-6,-11)(10,11,-1,-12)(12,9,-13,-10)(2,15,-3,-16)(16,3,-17,-4)(20,17,-15,-18)(7,18,-8,-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,-9,12)(-2,-16,-4,13,9)(-3,16)(-5,-11,10,-13)(-6,19,-8,1,11)(-7,-19)(-10,-12)(-14,5)(-15,2,8,18)(-17,20,6,14,4)(-18,7,-20)(3,15,17)
Multiloop annotated with half-edges
12^3_1 annotated with half-edges