$12^{1}_{354}$ - Minimal pinning sets
Pinning sets for 12^1_354 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_354 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 144
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: 2.97076
on average over minimal pinning sets: 2.31429
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, 4, 7, 11}
5
[2, 2, 2, 2, 3]
2.20
a (minimal)
• {1, 2, 3, 5, 6, 7, 11}
7
[2, 2, 2, 2, 3, 3, 3]
2.43
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
0
7
2.52
7
0
1
21
2.74
8
0
0
39
2.91
9
0
0
41
3.05
10
0
0
25
3.18
11
0
0
8
3.27
12
0
0
1
3.33
Total
1
1
142
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 3, 3, 5, 6, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,2],[0,1,5,3],[0,2,6,6],[0,7,7,1],[1,8,8,2],[3,9,9,3],[4,9,8,4],[5,7,9,5],[6,8,7,6]]
PD code (use to draw this loop with SnapPy): [[13,20,14,1],[12,7,13,8],[19,6,20,7],[14,6,15,5],[1,9,2,8],[18,11,19,12],[15,4,16,5],[9,3,10,2],[10,17,11,18],[3,16,4,17]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (11,20,-12,-1)(1,10,-2,-11)(15,2,-16,-3)(8,3,-9,-4)(4,17,-5,-18)(14,7,-15,-8)(19,12,-20,-13)(6,13,-7,-14)(9,16,-10,-17)(18,5,-19,-6)
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,-11)(-2,15,7,13,-20,11)(-3,8,-15)(-4,-18,-6,-14,-8)(-5,18)(-7,14)(-9,-17,4)(-10,1,-12,19,5,17)(-13,6,-19)(-16,9,3)(2,10,16)(12,20)
Loop annotated with half-edges
12^1_354 annotated with half-edges