$12^{1}_{351}$ - Minimal pinning sets
Pinning sets for 12^1_351
Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_351
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, 8, 11}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
•
{1, 2, 4, 7, 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: [[1,2,3,4],[0,4,5,6],[0,6,6,7],[0,7,7,4],[0,3,8,1],[1,9,9,6],[1,5,2,2],[2,8,3,3],[4,7,9,9],[5,8,8,5]]
PD code (use to draw this loop with SnapPy): [[20,9,1,10],[10,17,11,18],[6,19,7,20],[15,8,16,9],[1,16,2,17],[11,4,12,5],[18,5,19,6],[7,14,8,15],[2,14,3,13],[3,12,4,13]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (8,1,-9,-2)(15,2,-16,-3)(12,5,-13,-6)(17,6,-18,-7)(20,9,-1,-10)(10,19,-11,-20)(4,11,-5,-12)(18,13,-19,-14)(7,14,-8,-15)(3,16,-4,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,8,14,-19,10)(-2,15,-8)(-3,-17,-7,-15)(-4,-12,-6,17)(-5,12)(-9,20,-11,4,16,2)(-10,-20)(-13,18,6)(-14,7,-18)(-16,3)(1,9)(5,11,19,13)
Loop annotated with half-edges
12^1_351 annotated with half-edges