$12^{1}_{34}$ - Minimal pinning sets
Pinning sets for 12^1_34 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_34 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 128
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: 2.90623
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, 4, 5, 6}
5
[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
5
1
0
0
2.0
6
0
0
7
2.38
7
0
0
21
2.65
8
0
0
35
2.86
9
0
0
35
3.02
10
0
0
21
3.14
11
0
0
7
3.25
12
0
0
1
3.33
Total
1
0
127
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 2, 3, 3, 4, 4, 4, 5, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,5],[0,6,6,3],[0,2,7,8],[0,8,7,1],[1,6,6,1],[2,5,5,2],[3,4,9,9],[3,9,9,4],[7,8,8,7]]
PD code (use to draw this loop with SnapPy): [[20,15,1,16],[16,10,17,9],[19,6,20,7],[14,5,15,6],[1,11,2,10],[17,8,18,9],[7,18,8,19],[2,13,3,14],[4,11,5,12],[12,3,13,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (10,1,-11,-2)(12,3,-13,-4)(17,4,-18,-5)(15,6,-16,-7)(7,14,-8,-15)(8,19,-9,-20)(2,9,-3,-10)(20,11,-1,-12)(18,13,-19,-14)(5,16,-6,-17)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,10,-3,12)(-2,-10)(-4,17,-6,15,-8,-20,-12)(-5,-17)(-7,-15)(-9,2,-11,20)(-13,18,4)(-14,7,-16,5,-18)(-19,8,14)(1,11)(3,9,19,13)(6,16)
Loop annotated with half-edges
12^1_34 annotated with half-edges