$10^{1}_{21}$ - Minimal pinning sets
Pinning sets for 10^1_21 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 10^1_21 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 48
of which optimal: 2
of which minimal: 2
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.8307
on average over minimal pinning sets: 2.2
on average over optimal pinning sets: 2.2
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 4, 5, 9}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 2, 3, 5, 9}
5
[2, 2, 2, 2, 3]
2.20
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
2
0
0
2.2
6
0
0
9
2.56
7
0
0
16
2.8
8
0
0
14
2.98
9
0
0
6
3.11
10
0
0
1
3.2
Total
2
0
46
Other information about this loop 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: [[1,2,3,3],[0,4,4,2],[0,1,4,5],[0,5,6,0],[1,6,2,1],[2,7,7,3],[3,7,7,4],[5,6,6,5]]
PD code (use to draw this loop with SnapPy): [[7,16,8,1],[6,13,7,14],[15,12,16,13],[8,2,9,1],[14,5,15,6],[11,2,12,3],[9,4,10,5],[3,10,4,11]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (6,1,-7,-2)(11,2,-12,-3)(16,7,-1,-8)(12,9,-13,-10)(3,10,-4,-11)(4,13,-5,-14)(14,5,-15,-6)(8,15,-9,-16)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,6,-15,8)(-2,11,-4,-14,-6)(-3,-11)(-5,14)(-7,16,-9,12,2)(-8,-16)(-10,3,-12)(-13,4,10)(1,7)(5,13,9,15)
Loop annotated with half-edges
10^1_21 annotated with half-edges