$12^{1}_{163}$ - Minimal pinning sets
Pinning sets for 12^1_163 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_163 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, 3, 4, 11}
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],[0,5,5,2],[0,1,6,6],[0,7,7,4],[0,3,8,5],[1,4,9,1],[2,9,9,2],[3,8,8,3],[4,7,7,9],[5,8,6,6]]
PD code (use to draw this loop with SnapPy): [[20,5,1,6],[6,16,7,15],[19,14,20,15],[4,9,5,10],[1,9,2,8],[16,8,17,7],[13,18,14,19],[10,3,11,4],[2,11,3,12],[17,12,18,13]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (7,20,-8,-1)(17,2,-18,-3)(4,13,-5,-14)(5,18,-6,-19)(1,6,-2,-7)(19,8,-20,-9)(14,9,-15,-10)(16,11,-17,-12)(12,3,-13,-4)(10,15,-11,-16)
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,-7)(-2,17,11,15,9,-20,7)(-3,12,-17)(-4,-14,-10,-16,-12)(-5,-19,-9,14)(-6,1,-8,19)(-11,16)(-13,4)(-15,10)(-18,5,13,3)(2,6,18)(8,20)
Loop annotated with half-edges
12^1_163 annotated with half-edges