$12^{1}_{23}$ - Minimal pinning sets
Pinning sets for 12^1_23 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_23 Pinning data
Pinning number of this loop: 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, 3, 11}
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 loop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 6, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,4,5],[0,5,6,7],[0,8,8,5],[0,5,1,1],[1,4,3,2],[2,9,7,7],[2,6,6,9],[3,9,9,3],[6,8,8,7]]
PD code (use to draw this loop with SnapPy): [[3,20,4,1],[2,13,3,14],[16,19,17,20],[4,11,5,12],[1,15,2,14],[15,12,16,13],[7,18,8,19],[17,8,18,9],[10,5,11,6],[6,9,7,10]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (8,3,-9,-4)(15,6,-16,-7)(4,7,-5,-8)(2,9,-3,-10)(13,10,-14,-11)(20,11,-1,-12)(12,19,-13,-20)(5,16,-6,-17)(14,17,-15,-18)(1,18,-2,-19)
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,-19,12)(-2,-10,13,19)(-3,8,-5,-17,14,10)(-4,-8)(-6,15,17)(-7,4,-9,2,18,-15)(-11,20,-13)(-12,-20)(-14,-18,1,11)(-16,5,7)(3,9)(6,16)
Loop annotated with half-edges
12^1_23 annotated with half-edges