$12^{3}_{65}$ - Minimal pinning sets
Pinning sets for 12^3_65 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^3_65 Pinning data
Pinning number of this multiloop: 7
Total number of pinning sets: 32
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: 2.80821
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, 6, 7, 11}
7
[2, 2, 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
7
1
0
0
2.0
8
0
0
5
2.4
9
0
0
10
2.71
10
0
0
10
2.96
11
0
0
5
3.16
12
0
0
1
3.33
Total
1
0
31
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 2, 2, 2, 3, 3, 6, 6, 8]
Minimal region degree: 2
Is multisimple: Yes
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,4,4,2],[0,1,5,5],[0,6,6,0],[1,7,8,1],[2,9,9,2],[3,7,7,3],[4,6,6,8],[4,7,9,9],[5,8,8,5]]
PD code (use to draw this multiloop with SnapPy): [[8,14,1,9],[9,15,10,20],[7,19,8,20],[13,1,14,2],[15,11,16,10],[18,6,19,7],[2,12,3,13],[11,3,12,4],[16,4,17,5],[5,17,6,18]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (14,1,-9,-2)(15,2,-16,-3)(17,4,-18,-5)(10,7,-11,-8)(13,20,-14,-15)(8,9,-1,-10)(6,11,-7,-12)(3,16,-4,-17)(5,18,-6,-19)(19,12,-20,-13)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,14,20,12,-7,10)(-2,15,-14)(-3,-17,-5,-19,-13,-15)(-4,17)(-6,-12,19)(-8,-10)(-9,8,-11,6,18,4,16,2)(-16,3)(-18,5)(-20,13)(1,9)(7,11)
Multiloop annotated with half-edges
12^3_65 annotated with half-edges