$12^{2}_{33}$ - Minimal pinning sets
Pinning sets for 12^2_33 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_33 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, 6, 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, 4, 4, 4, 4, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,5,0],[0,5,3,3],[0,2,2,4],[1,3,6,7],[1,8,8,2],[4,8,9,9],[4,9,9,8],[5,7,6,5],[6,7,7,6]]
PD code (use to draw this multiloop with SnapPy): [[12,3,1,4],[4,11,5,12],[2,20,3,13],[1,20,2,19],[10,18,11,19],[5,14,6,13],[15,9,16,10],[17,7,18,8],[14,7,15,6],[8,16,9,17]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (10,1,-11,-2)(4,15,-5,-16)(18,5,-19,-6)(16,7,-17,-8)(2,9,-3,-10)(6,17,-7,-18)(8,19,-9,-20)(20,11,-13,-12)(12,13,-1,-14)(14,3,-15,-4)
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,-3,14)(-2,-10)(-4,-16,-8,-20,-12,-14)(-5,18,-7,16)(-6,-18)(-9,2,-11,20)(-13,12)(-15,4)(-17,6,-19,8)(1,13,11)(3,9,19,5,15)(7,17)
Multiloop annotated with half-edges
12^2_33 annotated with half-edges