$12^{1}_{247}$ - Minimal pinning sets
Pinning sets for 12^1_247 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_247 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, 3, 6, 9, 10}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 3, 6, 9, 11}
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: [[1,2,2,3],[0,4,5,5],[0,6,7,0],[0,8,8,4],[1,3,8,9],[1,9,6,1],[2,5,7,7],[2,6,6,9],[3,9,4,3],[4,8,7,5]]
PD code (use to draw this loop with SnapPy): [[15,20,16,1],[14,5,15,6],[19,16,20,17],[1,8,2,9],[6,9,7,10],[4,13,5,14],[17,13,18,12],[18,11,19,12],[7,2,8,3],[10,3,11,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (7,20,-8,-1)(11,2,-12,-3)(18,5,-19,-6)(3,6,-4,-7)(15,8,-16,-9)(9,14,-10,-15)(1,10,-2,-11)(16,13,-17,-14)(12,17,-13,-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,-11,-3,-7)(-2,11)(-4,-20,7)(-5,18,-13,16,8,20)(-6,3,-12,-18)(-8,15,-10,1)(-9,-15)(-14,9,-16)(-17,12,2,10,14)(-19,4,6)(5,19)(13,17)
Loop annotated with half-edges
12^1_247 annotated with half-edges