$12^{3}_{13}$ - Minimal pinning sets
Pinning sets for 12^3_13
Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^3_13
Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 256
of which optimal: 1
of which minimal: 1
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.96564
on average over minimal pinning sets: 2.0
on average over optimal pinning sets: 2.0
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
•
{1, 2, 6, 7}
4
[2, 2, 2, 2]
2.00
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
2.0
5
0
0
8
2.4
6
0
0
28
2.67
7
0
0
56
2.86
8
0
0
70
3.0
9
0
0
56
3.11
10
0
0
28
3.2
11
0
0
8
3.27
12
0
0
1
3.33
Total
1
0
255
Other information about this multiloop
Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 4, 4, 5, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,3,2,0],[0,1,4,5],[0,5,6,1],[2,6,7,7],[2,8,8,3],[3,9,9,4],[4,9,8,4],[5,7,9,5],[6,8,7,6]]
PD code (use to draw this multiloop with SnapPy): [[8,16,1,9],[9,7,10,8],[10,15,11,16],[1,6,2,7],[14,20,15,17],[11,5,12,6],[2,18,3,17],[19,13,20,14],[4,12,5,13],[18,4,19,3]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (1,10,-2,-11)(20,3,-13,-4)(7,4,-8,-5)(18,5,-19,-6)(2,13,-3,-14)(11,14,-12,-15)(19,16,-20,-17)(6,17,-7,-18)(15,12,-16,-9)(9,8,-10,-1)
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,-15,-9)(-2,-14,11)(-3,20,16,12,14)(-4,7,17,-20)(-5,18,-7)(-6,-18)(-8,9,-16,19,5)(-10,1)(-12,15)(-13,2,10,8,4)(-17,6,-19)(3,13)
Multiloop annotated with half-edges
12^3_13 annotated with half-edges