$12^{1}_{17}$ - Minimal pinning sets
Pinning sets for 12^1_17 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_17 Pinning data
Pinning number of this loop: 4
Total number of pinning sets: 320
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: 3.03463
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, 9, 11}
4
[2, 2, 2, 3]
2.25
a (minimal)
• {1, 2, 7, 10, 11}
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
8
2.56
6
0
0
34
2.77
7
0
0
71
2.94
8
0
0
90
3.06
9
0
0
71
3.16
10
0
0
34
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
1
1
318
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 4, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,6],[0,6,5,3],[0,2,4,4],[0,3,3,1],[1,2,7,8],[1,8,7,2],[5,6,9,9],[5,9,9,6],[7,8,8,7]]
PD code (use to draw this loop with SnapPy): [[20,3,1,4],[4,13,5,14],[10,19,11,20],[11,2,12,3],[1,12,2,13],[5,19,6,18],[14,9,15,10],[6,15,7,16],[8,17,9,18],[7,17,8,16]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (9,20,-10,-1)(18,1,-19,-2)(12,7,-13,-8)(19,10,-20,-11)(2,11,-3,-12)(4,13,-5,-14)(14,5,-15,-6)(6,15,-7,-16)(16,3,-17,-4)(8,17,-9,-18)
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)(-19,19)(-20,20)
Face permutation $\varphi=(\sigma\epsilon)^{-1}=$ (-1,18,-9)(-2,-12,-8,-18)(-3,16,-7,12)(-4,-14,-6,-16)(-5,14)(-10,19,1)(-11,2,-19)(-13,4,-17,8)(-15,6)(-20,9,17,3,11)(5,13,7,15)(10,20)
Loop annotated with half-edges
12^1_17 annotated with half-edges