$12^{3}_{59}$ - Minimal pinning sets
Pinning sets for 12^3_59 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^3_59 Pinning data
Pinning number of this multiloop: 6
Total number of pinning sets: 64
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: 2.85421
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, 4, 7, 12}
6
[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
6
1
0
0
2.0
7
0
0
6
2.38
8
0
0
15
2.67
9
0
0
20
2.89
10
0
0
15
3.07
11
0
0
6
3.21
12
0
0
1
3.33
Total
1
0
63
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 2, 2, 3, 4, 4, 4, 6, 7]
Minimal region degree: 2
Is multisimple: Yes
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,5,0],[0,5,5,3],[0,2,6,4],[1,3,7,7],[1,6,2,2],[3,5,8,8],[4,9,9,4],[6,9,9,6],[7,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[8,12,1,9],[9,7,10,8],[11,20,12,13],[1,20,2,19],[6,18,7,19],[10,14,11,13],[2,14,3,15],[17,5,18,6],[3,16,4,15],[4,16,5,17]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (12,3,-13,-4)(18,5,-19,-6)(6,17,-7,-18)(2,11,-3,-12)(4,13,-5,-14)(14,7,-15,-8)(20,15,-17,-16)(16,19,-9,-20)(8,9,-1,-10)(10,1,-11,-2)
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,-4,-14,-8,-10)(-3,12)(-5,18,-7,14)(-6,-18)(-9,8,-15,20)(-11,2)(-13,4)(-16,-20)(-17,6,-19,16)(1,9,19,5,13,3,11)(7,17,15)
Multiloop annotated with half-edges
12^3_59 annotated with half-edges