$12^{1}_{210}$ - Minimal pinning sets
Pinning sets for 12^1_210 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_210 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 264
of which optimal: 2
of which minimal: 6
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.04089
on average over minimal pinning sets: 2.52222
on average over optimal pinning sets: 2.4
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 4, 9, 11}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {1, 2, 3, 9, 11}
5
[2, 2, 2, 3, 3]
2.40
a (minimal)
• {1, 2, 4, 7, 10, 11}
6
[2, 2, 2, 3, 3, 4]
2.67
b (minimal)
• {1, 2, 4, 5, 10, 11}
6
[2, 2, 2, 3, 3, 3]
2.50
c (minimal)
• {1, 2, 3, 7, 10, 11}
6
[2, 2, 2, 3, 3, 4]
2.67
d (minimal)
• {1, 2, 3, 5, 10, 11}
6
[2, 2, 2, 3, 3, 3]
2.50
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
2
0
0
2.4
6
0
4
13
2.66
7
0
0
52
2.87
8
0
0
80
3.04
9
0
0
69
3.15
10
0
0
34
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
2
4
258
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,2],[0,1,5,6],[0,7,7,4],[0,3,8,5],[1,4,2,1],[2,8,9,9],[3,9,8,3],[4,7,9,6],[6,8,7,6]]
PD code (use to draw this loop with SnapPy): [[11,20,12,1],[10,17,11,18],[19,16,20,17],[12,7,13,8],[1,8,2,9],[18,9,19,10],[15,4,16,5],[6,13,7,14],[2,6,3,5],[3,14,4,15]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (11,20,-12,-1)(1,18,-2,-19)(8,3,-9,-4)(17,4,-18,-5)(14,5,-15,-6)(2,9,-3,-10)(19,10,-20,-11)(15,12,-16,-13)(6,13,-7,-14)(7,16,-8,-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)(-19,19)(-20,20)
Face permutation $\varphi=(\sigma\epsilon)^{-1}=$ (-1,-19,-11)(-2,-10,19)(-3,8,16,12,20,10)(-4,17,-8)(-5,14,-7,-17)(-6,-14)(-9,2,18,4)(-12,15,5,-18,1)(-13,6,-15)(-16,7,13)(-20,11)(3,9)
Loop annotated with half-edges
12^1_210 annotated with half-edges