$11^{1}_{23}$ - Minimal pinning sets
Pinning sets for 11^1_23 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^1_23 Pinning data
Pinning number of this loop: 4
Total number of pinning sets: 160
of which optimal: 1
of which minimal: 2
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.97092
on average over minimal pinning sets: 2.325
on average over optimal pinning sets: 2.25
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 6, 10}
4
[2, 2, 2, 3]
2.25
a (minimal)
• {1, 4, 5, 6, 10}
5
[2, 2, 2, 3, 3]
2.40
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
1
0
0
2.25
5
0
1
7
2.55
6
0
0
26
2.77
7
0
0
45
2.93
8
0
0
45
3.06
9
0
0
26
3.15
10
0
0
8
3.23
11
0
0
1
3.27
Total
1
1
158
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 3, 4, 5, 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],[1,3,7,5],[2,4,8,6],[2,5,8,7],[4,6,8,8],[5,7,7,6]]
PD code (use to draw this loop with SnapPy): [[3,18,4,1],[2,9,3,10],[17,4,18,5],[1,11,2,10],[11,8,12,9],[5,12,6,13],[13,16,14,17],[14,7,15,8],[6,15,7,16]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (15,2,-16,-3)(12,5,-13,-6)(9,6,-10,-7)(18,7,-1,-8)(8,17,-9,-18)(3,10,-4,-11)(4,13,-5,-14)(11,14,-12,-15)(1,16,-2,-17)
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)
Face permutation $\varphi=(\sigma\epsilon)^{-1}=$ (-1,-17,8)(-2,15,-12,-6,9,17)(-3,-11,-15)(-4,-14,11)(-5,12,14)(-7,18,-9)(-8,-18)(-10,3,-16,1,7)(-13,4,10,6)(2,16)(5,13)
Loop annotated with half-edges
11^1_23 annotated with half-edges