$12^{3}_{14}$ - Minimal pinning sets
Pinning sets for 12^3_14
Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^3_14
Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 256
of which optimal: 3
of which minimal: 3
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.0346
on average over minimal pinning sets: 2.4
on average over optimal pinning sets: 2.4
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
•
{1, 3, 4, 7, 11}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
•
{1, 2, 4, 7, 11}
5
[2, 2, 2, 3, 3]
2.40
C (optimal)
•
{2, 4, 5, 7, 11}
5
[2, 2, 2, 3, 3]
2.40
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
3
0
0
2.4
6
0
0
19
2.68
7
0
0
51
2.89
8
0
0
75
3.03
9
0
0
65
3.15
10
0
0
33
3.23
11
0
0
9
3.29
12
0
0
1
3.33
Total
3
0
253
Other information about this multiloop
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: [[1,1,2,3],[0,3,4,0],[0,4,4,5],[0,5,6,1],[1,7,2,2],[2,8,6,3],[3,5,8,9],[4,9,9,8],[5,7,9,6],[6,8,7,7]]
PD code (use to draw this multiloop with SnapPy): [[10,14,1,11],[11,9,12,10],[13,20,14,15],[1,8,2,9],[12,16,13,15],[7,19,8,20],[2,19,3,18],[16,5,17,4],[6,3,7,4],[17,5,18,6]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (4,1,-5,-2)(2,15,-3,-16)(16,3,-11,-4)(9,6,-10,-7)(18,7,-19,-8)(8,17,-9,-18)(11,10,-12,-1)(5,12,-6,-13)(20,13,-17,-14)(14,19,-15,-20)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,4,-11)(-2,-16,-4)(-3,16)(-5,-13,20,-15,2)(-6,9,17,13)(-7,18,-9)(-8,-18)(-10,11,3,15,19,7)(-12,5,1)(-14,-20)(-17,8,-19,14)(6,12,10)
Multiloop annotated with half-edges
12^3_14 annotated with half-edges