$12^{1}_{349}$ - Minimal pinning sets
Pinning sets for 12^1_349 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_349 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 160
of which optimal: 1
of which minimal: 2
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.97043
on average over minimal pinning sets: 2.26667
on average over optimal pinning sets: 2.2
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 5, 9}
5
[2, 2, 2, 2, 3]
2.20
a (minimal)
• {1, 2, 3, 5, 7, 10}
6
[2, 2, 2, 2, 3, 3]
2.33
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
2.2
6
0
1
7
2.5
7
0
0
26
2.74
8
0
0
45
2.92
9
0
0
45
3.07
10
0
0
26
3.18
11
0
0
8
3.27
12
0
0
1
3.33
Total
1
1
158
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,5],[0,6,6,3],[0,2,7,4],[0,3,8,1],[1,9,9,1],[2,9,7,2],[3,6,8,8],[4,7,7,9],[5,8,6,5]]
PD code (use to draw this loop with SnapPy): [[20,5,1,6],[6,15,7,16],[12,19,13,20],[13,4,14,5],[1,14,2,15],[7,17,8,16],[18,11,19,12],[3,10,4,11],[2,10,3,9],[17,9,18,8]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (11,2,-12,-3)(18,3,-19,-4)(6,15,-7,-16)(7,20,-8,-1)(1,8,-2,-9)(16,9,-17,-10)(19,12,-20,-13)(4,13,-5,-14)(14,5,-15,-6)(10,17,-11,-18)
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,-9,16,-7)(-2,11,17,9)(-3,18,-11)(-4,-14,-6,-16,-10,-18)(-5,14)(-8,1)(-12,19,3)(-13,4,-19)(-15,6)(-17,10)(-20,7,15,5,13)(2,8,20,12)
Loop annotated with half-edges
12^1_349 annotated with half-edges