$12^{1}_{258}$ - Minimal pinning sets
Pinning sets for 12^1_258 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_258 Pinning data
Pinning number of this loop: 6
Total number of pinning sets: 128
of which optimal: 3
of which minimal: 3
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.97249
on average over minimal pinning sets: 2.33333
on average over optimal pinning sets: 2.33333
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 4, 5, 6, 9, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
B (optimal)
• {1, 3, 5, 6, 9, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
C (optimal)
• {2, 3, 5, 6, 9, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
6
3
0
0
2.33
7
0
0
16
2.64
8
0
0
35
2.87
9
0
0
40
3.04
10
0
0
25
3.18
11
0
0
8
3.27
12
0
0
1
3.33
Total
3
0
125
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 6, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,4,5,6],[0,6,6,0],[0,7,8,8],[1,9,7,5],[1,4,7,6],[1,5,2,2],[3,5,4,9],[3,9,9,3],[4,8,8,7]]
PD code (use to draw this loop with SnapPy): [[7,20,8,1],[17,6,18,7],[19,8,20,9],[1,15,2,14],[16,11,17,12],[5,10,6,11],[18,10,19,9],[15,5,16,4],[2,13,3,14],[12,3,13,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (9,20,-10,-1)(1,12,-2,-13)(14,3,-15,-4)(16,5,-17,-6)(6,15,-7,-16)(2,7,-3,-8)(13,8,-14,-9)(19,10,-20,-11)(11,18,-12,-19)(4,17,-5,-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,-13,-9)(-2,-8,13)(-3,14,8)(-4,-18,11,-20,9,-14)(-5,16,-7,2,12,18)(-6,-16)(-10,19,-12,1)(-11,-19)(-15,6,-17,4)(3,7,15)(5,17)(10,20)
Loop annotated with half-edges
12^1_258 annotated with half-edges