$12^{1}_{44}$ - Minimal pinning sets
Pinning sets for 12^1_44
Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_44
Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 160
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: 2.97043
on average over minimal pinning sets: 2.26667
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)
•
{2, 4, 5, 11, 12}
5
[2, 2, 2, 2, 3]
2.20
a (minimal)
•
{1, 2, 4, 5, 9, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
1
0
0
2.2
6
0
1
7
2.5
7
0
0
26
2.74
8
0
0
45
2.92
9
0
0
45
3.07
10
0
0
26
3.18
11
0
0
8
3.27
12
0
0
1
3.33
Total
1
1
158
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],[0,7,7,8],[0,8,8,6],[0,5,5,1],[1,4,4,6],[1,5,3,9],[2,9,9,2],[2,9,3,3],[6,8,7,7]]
PD code (use to draw this loop with SnapPy): [[20,9,1,10],[10,13,11,14],[6,19,7,20],[8,15,9,16],[1,12,2,13],[11,2,12,3],[14,3,15,4],[18,5,19,6],[7,17,8,16],[4,17,5,18]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (9,20,-10,-1)(12,1,-13,-2)(16,3,-17,-4)(18,7,-19,-8)(19,10,-20,-11)(8,11,-9,-12)(6,13,-7,-14)(14,5,-15,-6)(2,15,-3,-16)(4,17,-5,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,12,-9)(-2,-16,-4,-18,-8,-12)(-3,16)(-5,14,-7,18)(-6,-14)(-10,19,7,13,1)(-11,8,-19)(-13,6,-15,2)(-17,4)(-20,9,11)(3,15,5,17)(10,20)
Loop annotated with half-edges
12^1_44 annotated with half-edges