$12^{1}_{202}$ - Minimal pinning sets
Pinning sets for 12^1_202 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_202 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 272
of which optimal: 3
of which minimal: 4
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.0386
on average over minimal pinning sets: 2.46667
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, 3, 5, 7, 11}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {1, 3, 5, 6, 11}
5
[2, 2, 2, 3, 3]
2.40
C (optimal)
• {1, 2, 3, 5, 7}
5
[2, 2, 2, 3, 3]
2.40
a (minimal)
• {1, 2, 3, 5, 6, 9}
6
[2, 2, 2, 3, 3, 4]
2.67
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
3
0
0
2.4
6
0
1
19
2.68
7
0
0
55
2.89
8
0
0
81
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
3
1
268
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 3, 4, 5, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,2,0],[0,1,5,3],[0,2,6,7],[1,7,7,8],[2,8,9,6],[3,5,9,7],[3,6,4,4],[4,9,9,5],[5,8,8,6]]
PD code (use to draw this loop with SnapPy): [[20,13,1,14],[14,19,15,20],[15,12,16,13],[1,16,2,17],[18,5,19,6],[11,8,12,9],[2,8,3,7],[17,7,18,6],[4,9,5,10],[10,3,11,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (7,20,-8,-1)(15,2,-16,-3)(3,14,-4,-15)(4,19,-5,-20)(8,5,-9,-6)(1,6,-2,-7)(12,9,-13,-10)(17,10,-18,-11)(18,13,-19,-14)(11,16,-12,-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,-7)(-2,15,-4,-20,7)(-3,-15)(-5,8,20)(-6,1,-8)(-9,12,16,2,6)(-10,17,-12)(-11,-17)(-13,18,10)(-14,3,-16,11,-18)(-19,4,14)(5,19,13,9)
Loop annotated with half-edges
12^1_202 annotated with half-edges