$10^{1}_{23}$ - Minimal pinning sets
Pinning sets for 10^1_23 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 10^1_23 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, 3, 6, 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, 3, 4, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,4,5,5],[0,5,6,3],[0,2,6,0],[1,7,7,5],[1,4,2,1],[2,7,7,3],[4,6,6,4]]
PD code (use to draw this loop with SnapPy): [[5,16,6,1],[4,11,5,12],[15,2,16,3],[6,2,7,1],[12,9,13,10],[10,3,11,4],[14,7,15,8],[8,13,9,14]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (10,3,-11,-4)(8,5,-9,-6)(16,7,-1,-8)(4,9,-5,-10)(1,12,-2,-13)(13,2,-14,-3)(11,14,-12,-15)(6,15,-7,-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,-13,-3,10,-5,8)(-2,13)(-4,-10)(-6,-16,-8)(-7,16)(-9,4,-11,-15,6)(-12,1,7,15)(-14,11,3)(2,12,14)(5,9)
Loop annotated with half-edges
10^1_23 annotated with half-edges