$12^{1}_{10}$ - Minimal pinning sets
Pinning sets for 12^1_10 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_10 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, 5, 9}
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, 3, 4, 5, 6, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,4,5],[0,6,6,7],[0,8,8,5],[0,5,1,1],[1,4,3,6],[2,5,7,2],[2,6,9,9],[3,9,9,3],[7,8,8,7]]
PD code (use to draw this loop with SnapPy): [[3,20,4,1],[2,15,3,16],[19,8,20,9],[4,13,5,14],[1,17,2,16],[17,14,18,15],[9,18,10,19],[10,7,11,8],[12,5,13,6],[6,11,7,12]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (9,2,-10,-3)(7,4,-8,-5)(16,5,-17,-6)(3,8,-4,-9)(1,10,-2,-11)(14,11,-15,-12)(19,12,-20,-13)(13,18,-14,-19)(6,15,-7,-16)(20,17,-1,-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,-11,14,18)(-2,9,-4,7,15,11)(-3,-9)(-5,16,-7)(-6,-16)(-8,3,-10,1,17,5)(-12,19,-14)(-13,-19)(-15,6,-17,20,12)(-18,13,-20)(2,10)(4,8)
Loop annotated with half-edges
12^1_10 annotated with half-edges