$12^{1}_{167}$ - Minimal pinning sets
Pinning sets for 12^1_167 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_167 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 192
of which optimal: 2
of which minimal: 2
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.96906
on average over minimal pinning sets: 2.2
on average over optimal pinning sets: 2.2
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 4, 7, 9}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 2, 4, 5, 9}
5
[2, 2, 2, 2, 3]
2.20
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
2.2
6
0
0
13
2.54
7
0
0
36
2.78
8
0
0
55
2.95
9
0
0
50
3.09
10
0
0
27
3.19
11
0
0
8
3.27
12
0
0
1
3.33
Total
2
0
190
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 4, 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],[0,6,7,7],[0,8,8,5],[1,4,2,1],[2,8,9,3],[3,9,9,3],[4,9,6,4],[6,8,7,7]]
PD code (use to draw this loop with SnapPy): [[11,20,12,1],[10,17,11,18],[19,16,20,17],[12,6,13,5],[1,8,2,9],[18,9,19,10],[15,6,16,7],[13,4,14,5],[7,2,8,3],[3,14,4,15]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (10,1,-11,-2)(2,9,-3,-10)(12,3,-13,-4)(18,5,-19,-6)(15,6,-16,-7)(20,11,-1,-12)(16,13,-17,-14)(7,14,-8,-15)(8,17,-9,-18)(4,19,-5,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,10,-3,12)(-2,-10)(-4,-20,-12)(-5,18,-9,2,-11,20)(-6,15,-8,-18)(-7,-15)(-13,16,6,-19,4)(-14,7,-16)(-17,8,14)(1,11)(3,9,17,13)(5,19)
Loop annotated with half-edges
12^1_167 annotated with half-edges