$11^{1}_{33}$ - Minimal pinning sets
Pinning sets for 11^1_33 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^1_33 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, 2, 4, 10}
4
[2, 2, 2, 3]
2.25
a (minimal)
• {1, 2, 3, 5, 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: [[1,2,3,4],[0,4,5,5],[0,6,7,3],[0,2,7,7],[0,6,8,1],[1,8,8,1],[2,8,4,7],[2,6,3,3],[4,6,5,5]]
PD code (use to draw this loop with SnapPy): [[5,18,6,1],[4,11,5,12],[14,17,15,18],[6,15,7,16],[1,13,2,12],[10,3,11,4],[13,8,14,9],[16,7,17,8],[2,9,3,10]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (7,18,-8,-1)(13,4,-14,-5)(10,5,-11,-6)(17,6,-18,-7)(1,8,-2,-9)(9,16,-10,-17)(3,12,-4,-13)(11,14,-12,-15)(2,15,-3,-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)(-17,17)(-18,18)
Face permutation $\varphi=(\sigma\epsilon)^{-1}=$ (-1,-9,-17,-7)(-2,-16,9)(-3,-13,-5,10,16)(-4,13)(-6,17,-10)(-8,1)(-11,-15,2,8,18,6)(-12,3,15)(-14,11,5)(-18,7)(4,12,14)
Loop annotated with half-edges
11^1_33 annotated with half-edges