$12^{2}_{6}$ - Minimal pinning sets
Pinning sets for 12^2_6 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_6 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, 3, 5, 6, 7, 11}
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, 3, 5, 5, 6, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,3,3,4],[0,5,6,0],[0,4,1,1],[1,3,7,7],[2,7,8,8],[2,9,9,7],[4,6,5,4],[5,9,9,5],[6,8,8,6]]
PD code (use to draw this multiloop with SnapPy): [[3,10,4,1],[2,20,3,11],[9,4,10,5],[1,12,2,11],[12,19,13,20],[5,18,6,17],[8,14,9,15],[18,13,19,14],[6,16,7,17],[15,7,16,8]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (7,2,-8,-3)(16,5,-17,-6)(1,8,-2,-9)(19,14,-20,-15)(6,15,-7,-16)(4,17,-5,-18)(18,3,-19,-4)(13,20,-14,-11)(10,11,-1,-12)(12,9,-13,-10)
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)(-2,7,15,-20,13,9)(-3,18,-5,16,-7)(-4,-18)(-6,-16)(-8,1,11,-14,19,3)(-10,-12)(-11,10,-13)(-15,6,-17,4,-19)(2,8)(5,17)(14,20)
Multiloop annotated with half-edges
12^2_6 annotated with half-edges