$12^{3}_{61}$ - Minimal pinning sets
Pinning sets for 12^3_61 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^3_61 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, 5, 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, 4, 4, 5, 6, 6]
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],[1,7,5,1],[2,4,8,2],[3,8,7,3],[4,6,9,9],[5,9,9,6],[7,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[8,16,1,9],[9,17,10,20],[7,19,8,20],[15,1,16,2],[17,11,18,10],[18,6,19,7],[2,14,3,15],[11,3,12,4],[13,5,14,6],[12,5,13,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (14,1,-15,-2)(2,15,-3,-16)(16,3,-9,-4)(17,4,-18,-5)(10,7,-11,-8)(8,9,-1,-10)(6,11,-7,-12)(13,20,-14,-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,-16,-4,17,-14)(-3,16)(-5,-19,-13,-17)(-6,-12,19)(-8,-10)(-9,8,-11,6,18,4)(-15,2)(-18,5)(-20,13)(1,9,3,15)(7,11)
Multiloop annotated with half-edges
12^3_61 annotated with half-edges