$12^{1}_{325}$ - Minimal pinning sets
Pinning sets for 12^1_325
Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_325
Pinning data
Pinning number of this loop: 4
Total number of pinning sets: 288
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: 3.03457
on average over minimal pinning sets: 2.375
on average over optimal pinning sets: 2.25
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
•
{1, 4, 7, 12}
4
[2, 2, 2, 3]
2.25
a (minimal)
•
{1, 2, 4, 7, 9, 11}
6
[2, 2, 2, 3, 3, 3]
2.50
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
1
0
0
2.25
5
0
0
8
2.57
6
0
1
28
2.78
7
0
0
61
2.93
8
0
0
80
3.05
9
0
0
66
3.15
10
0
0
33
3.23
11
0
0
9
3.29
12
0
0
1
3.33
Total
1
1
286
Other information about this loop
Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,3,4,5],[0,5,6,0],[0,7,4,1],[1,3,7,5],[1,4,8,2],[2,8,8,9],[3,9,9,4],[5,9,6,6],[6,8,7,7]]
PD code (use to draw this loop with SnapPy): [[20,5,1,6],[6,15,7,16],[4,19,5,20],[1,14,2,15],[7,2,8,3],[16,3,17,4],[11,18,12,19],[13,8,14,9],[17,10,18,11],[12,10,13,9]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (5,20,-6,-1)(14,1,-15,-2)(18,3,-19,-4)(11,6,-12,-7)(7,10,-8,-11)(15,8,-16,-9)(19,12,-20,-13)(4,13,-5,-14)(9,16,-10,-17)(2,17,-3,-18)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,14,-5)(-2,-18,-4,-14)(-3,18)(-6,11,-8,15,1)(-7,-11)(-9,-17,2,-15)(-10,7,-12,19,3,17)(-13,4,-19)(-16,9)(-20,5,13)(6,20,12)(8,10,16)
Loop annotated with half-edges
12^1_325 annotated with half-edges