$12^{1}_{348}$ - Minimal pinning sets
Pinning sets for 12^1_348 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_348 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, 2, 3, 7, 10, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
B (optimal)
• {1, 2, 3, 7, 8, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
C (optimal)
• {1, 2, 3, 5, 8, 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: [[1,2,3,4],[0,4,5,6],[0,6,6,3],[0,2,7,4],[0,3,5,1],[1,4,8,8],[1,7,2,2],[3,6,9,9],[5,9,9,5],[7,8,8,7]]
PD code (use to draw this loop with SnapPy): [[20,7,1,8],[8,17,9,18],[14,19,15,20],[15,6,16,7],[1,16,2,17],[9,2,10,3],[18,13,19,14],[5,12,6,13],[10,4,11,3],[11,4,12,5]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (20,7,-1,-8)(8,1,-9,-2)(12,3,-13,-4)(17,4,-18,-5)(2,9,-3,-10)(15,10,-16,-11)(18,13,-19,-14)(5,14,-6,-15)(11,16,-12,-17)(6,19,-7,-20)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,8)(-2,-10,15,-6,-20,-8)(-3,12,16,10)(-4,17,-12)(-5,-15,-11,-17)(-7,20)(-9,2)(-13,18,4)(-14,5,-18)(-16,11)(-19,6,14)(1,7,19,13,3,9)
Loop annotated with half-edges
12^1_348 annotated with half-edges