$12^{1}_{141}$ - Minimal pinning sets
Pinning sets for 12^1_141 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_141 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 176
of which optimal: 1
of which minimal: 3
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.97631
on average over minimal pinning sets: 2.34444
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, 4, 5, 7, 11}
5
[2, 2, 2, 2, 3]
2.20
a (minimal)
• {1, 2, 4, 6, 7, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
b (minimal)
• {1, 3, 4, 6, 7, 11}
6
[2, 2, 2, 2, 3, 4]
2.50
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
2
7
2.5
7
0
0
30
2.75
8
0
0
51
2.94
9
0
0
49
3.08
10
0
0
27
3.19
11
0
0
8
3.27
12
0
0
1
3.33
Total
1
2
173
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: [[1,1,2,3],[0,3,4,0],[0,4,5,5],[0,6,6,1],[1,7,5,2],[2,4,8,2],[3,8,9,3],[4,9,9,8],[5,7,9,6],[6,8,7,7]]
PD code (use to draw this loop with SnapPy): [[20,9,1,10],[10,19,11,20],[8,13,9,14],[1,18,2,19],[11,15,12,14],[12,7,13,8],[17,2,18,3],[15,5,16,4],[6,3,7,4],[16,5,17,6]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (15,20,-16,-1)(4,1,-5,-2)(2,13,-3,-14)(14,3,-15,-4)(10,5,-11,-6)(18,7,-19,-8)(6,11,-7,-12)(12,9,-13,-10)(19,16,-20,-17)(8,17,-9,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,4,-15)(-2,-14,-4)(-3,14)(-5,10,-13,2)(-6,-12,-10)(-7,18,-9,12)(-8,-18)(-11,6)(-16,19,7,11,5,1)(-17,8,-19)(-20,15,3,13,9,17)(16,20)
Loop annotated with half-edges
12^1_141 annotated with half-edges