$12^{1}_{3}$ - Minimal pinning sets
Pinning sets for 12^1_3 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_3 Pinning data
Pinning number of this loop: 6
Total number of pinning sets: 64
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: 2.85421
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, 3, 4, 7, 9}
6
[2, 2, 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
6
1
0
0
2.0
7
0
0
6
2.38
8
0
0
15
2.67
9
0
0
20
2.89
10
0
0
15
3.07
11
0
0
6
3.21
12
0
0
1
3.33
Total
1
0
63
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 2, 2, 3, 3, 5, 5, 6, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,3,4,5],[0,6,6,3],[0,2,1,0],[1,6,7,7],[1,8,8,6],[2,5,4,2],[4,9,9,4],[5,9,9,5],[7,8,8,7]]
PD code (use to draw this loop with SnapPy): [[20,11,1,12],[12,10,13,9],[19,2,20,3],[10,1,11,2],[13,4,14,5],[17,8,18,9],[3,18,4,19],[14,6,15,5],[7,16,8,17],[6,16,7,15]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (17,20,-18,-1)(11,2,-12,-3)(3,12,-4,-13)(15,6,-16,-7)(7,10,-8,-11)(18,9,-19,-10)(13,4,-14,-5)(5,14,-6,-15)(1,16,-2,-17)(8,19,-9,-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,-17)(-2,11,-8,-20,17)(-3,-13,-5,-15,-7,-11)(-4,13)(-6,15)(-9,18,20)(-10,7,-16,1,-18)(-12,3)(-14,5)(-19,8,10)(2,16,6,14,4,12)(9,19)
Loop annotated with half-edges
12^1_3 annotated with half-edges