$10^{2}_{12}$ - Minimal pinning sets
Pinning sets for 10^2_12 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 10^2_12 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 60
of which optimal: 4
of which minimal: 4
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.83683
on average over minimal pinning sets: 2.3
on average over optimal pinning sets: 2.3
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 4, 6, 7}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 3, 5, 6, 7}
5
[2, 2, 2, 2, 4]
2.40
C (optimal)
• {1, 2, 3, 6, 7}
5
[2, 2, 2, 2, 3]
2.20
D (optimal)
• {1, 3, 6, 7, 9}
5
[2, 2, 2, 2, 4]
2.40
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
4
0
0
2.3
6
0
0
14
2.64
7
0
0
20
2.86
8
0
0
15
3.0
9
0
0
6
3.11
10
0
0
1
3.2
Total
4
0
56
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,3,4,0],[0,5,5,6],[0,6,7,1],[1,7,5,5],[2,4,4,2],[2,7,7,3],[3,6,6,4]]
PD code (use to draw this multiloop with SnapPy): [[6,16,1,7],[7,5,8,6],[10,15,11,16],[1,4,2,5],[8,13,9,14],[14,9,15,10],[11,3,12,4],[2,12,3,13]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (16,1,-7,-2)(5,2,-6,-3)(12,3,-13,-4)(9,14,-10,-15)(6,7,-1,-8)(13,8,-14,-9)(15,10,-16,-11)(4,11,-5,-12)
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)
Face permutation $\varphi=(\sigma\epsilon)^{-1}=$ (-1,16,10,14,8)(-2,5,11,-16)(-3,12,-5)(-4,-12)(-6,-8,13,3)(-7,6,2)(-9,-15,-11,4,-13)(-10,15)(-14,9)(1,7)
Multiloop annotated with half-edges
10^2_12 annotated with half-edges