$12^{1}_{279}$ - Minimal pinning sets
Pinning sets for 12^1_279
Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_279
Pinning data
Pinning number of this loop: 6
Total number of pinning sets: 96
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: 2.91189
on average over minimal pinning sets: 2.16667
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, 4, 5, 6, 9, 11}
6
[2, 2, 2, 2, 2, 3]
2.17
B (optimal)
•
{1, 3, 5, 6, 9, 11}
6
[2, 2, 2, 2, 2, 3]
2.17
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
6
2
0
0
2.17
7
0
0
11
2.52
8
0
0
25
2.78
9
0
0
30
2.98
10
0
0
20
3.13
11
0
0
7
3.25
12
0
0
1
3.33
Total
2
0
94
Other information about this loop
Properties
Region degree sequence: [2, 2, 2, 2, 2, 3, 3, 3, 4, 4, 5, 8]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,4,4,2],[0,1,4,5],[0,6,6,0],[1,7,2,1],[2,7,8,6],[3,5,8,3],[4,9,9,5],[5,9,9,6],[7,8,8,7]]
PD code (use to draw this loop with SnapPy): [[20,9,1,10],[10,18,11,17],[19,16,20,17],[8,1,9,2],[18,12,19,11],[6,15,7,16],[2,7,3,8],[12,5,13,6],[14,3,15,4],[4,13,5,14]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (7,20,-8,-1)(17,2,-18,-3)(15,4,-16,-5)(6,13,-7,-14)(19,8,-20,-9)(1,10,-2,-11)(14,11,-15,-12)(12,5,-13,-6)(3,16,-4,-17)(9,18,-10,-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,-11,14,-7)(-2,17,-4,15,11)(-3,-17)(-5,12,-15)(-6,-14,-12)(-8,19,-10,1)(-9,-19)(-13,6)(-16,3,-18,9,-20,7,13,5)(2,10,18)(4,16)(8,20)
Loop annotated with half-edges
12^1_279 annotated with half-edges