$12^{1}_{113}$ - Minimal pinning sets
Pinning sets for 12^1_113 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_113 Pinning data
Pinning number of this loop: 4
Total number of pinning sets: 384
of which optimal: 2
of which minimal: 2
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.03466
on average over minimal pinning sets: 2.25
on average over optimal pinning sets: 2.25
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 5, 11}
4
[2, 2, 2, 3]
2.25
B (optimal)
• {1, 2, 5, 11}
4
[2, 2, 2, 3]
2.25
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
2
0
0
2.25
5
0
0
15
2.59
6
0
0
49
2.81
7
0
0
91
2.97
8
0
0
105
3.08
9
0
0
77
3.17
10
0
0
35
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
2
0
382
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,4,5],[0,5,6,6],[0,6,7,8],[0,5,1,1],[1,4,8,2],[2,9,3,2],[3,9,9,8],[3,7,9,5],[6,8,7,7]]
PD code (use to draw this loop with SnapPy): [[3,20,4,1],[11,2,12,3],[19,8,20,9],[4,18,5,17],[1,10,2,11],[12,10,13,9],[7,18,8,19],[5,15,6,14],[16,13,17,14],[6,15,7,16]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (9,20,-10,-1)(17,6,-18,-7)(7,10,-8,-11)(19,8,-20,-9)(14,11,-15,-12)(12,3,-13,-4)(4,13,-5,-14)(15,2,-16,-3)(5,16,-6,-17)(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,-9)(-2,15,11,-8,19)(-3,12,-15)(-4,-14,-12)(-5,-17,-7,-11,14)(-6,17)(-10,7,-18,1)(-13,4)(-16,5,13,3)(-20,9)(2,18,6,16)(8,10,20)
Loop annotated with half-edges
12^1_113 annotated with half-edges