$12^{1}_{121}$ - Minimal pinning sets
Pinning sets for 12^1_121 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_121 Pinning data
Pinning number of this loop: 6
Total number of pinning sets: 80
of which optimal: 1
of which minimal: 2
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.91429
on average over minimal pinning sets: 2.22619
on average over optimal pinning sets: 2.16667
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 4, 6, 11}
6
[2, 2, 2, 2, 2, 3]
2.17
a (minimal)
• {1, 2, 3, 4, 5, 8, 11}
7
[2, 2, 2, 2, 2, 3, 3]
2.29
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
6
1
0
0
2.17
7
0
1
6
2.47
8
0
0
19
2.73
9
0
0
26
2.94
10
0
0
19
3.12
11
0
0
7
3.25
12
0
0
1
3.33
Total
1
1
78
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 7, 8]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,4,5],[0,6,7,3],[0,2,7,7],[0,5,1,1],[1,4,8,8],[2,9,9,7],[2,6,3,3],[5,9,9,5],[6,8,8,6]]
PD code (use to draw this loop with SnapPy): [[3,20,4,1],[11,2,12,3],[16,19,17,20],[4,17,5,18],[1,10,2,11],[12,10,13,9],[15,6,16,7],[18,5,19,6],[13,8,14,9],[7,14,8,15]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (9,20,-10,-1)(16,3,-17,-4)(13,4,-14,-5)(11,6,-12,-7)(7,10,-8,-11)(19,8,-20,-9)(5,12,-6,-13)(2,15,-3,-16)(14,17,-15,-18)(1,18,-2,-19)
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,-9)(-2,-16,-4,13,-6,11,-8,19)(-3,16)(-5,-13)(-7,-11)(-10,7,-12,5,-14,-18,1)(-15,2,18)(-17,14,4)(-20,9)(3,15,17)(6,12)(8,10,20)
Loop annotated with half-edges
12^1_121 annotated with half-edges