$10^{3}_{4}$ - Minimal pinning sets
Pinning sets for 10^3_4 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 10^3_4 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 64
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: 2.8189
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, 3, 4, 7}
4
[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
4
1
0
0
2.0
5
0
0
6
2.4
6
0
0
15
2.67
7
0
0
20
2.86
8
0
0
15
3.0
9
0
0
6
3.11
10
0
0
1
3.2
Total
1
0
63
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 4, 4, 4, 4, 4, 4]
Minimal region degree: 2
Is multisimple: Yes
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,4,5,5],[0,5,6,0],[0,6,4,4],[1,3,3,7],[1,7,2,1],[2,7,7,3],[4,6,6,5]]
PD code (use to draw this multiloop with SnapPy): [[8,12,1,9],[9,3,10,4],[11,7,12,8],[1,13,2,16],[2,15,3,16],[10,5,11,4],[6,13,7,14],[14,5,15,6]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (16,1,-13,-2)(2,13,-3,-14)(12,3,-9,-4)(14,5,-15,-6)(6,15,-7,-16)(8,9,-1,-10)(10,7,-11,-8)(4,11,-5,-12)
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)
Face permutation $\varphi=(\sigma\epsilon)^{-1}=$ (-1,16,-7,10)(-2,-14,-6,-16)(-3,12,-5,14)(-4,-12)(-8,-10)(-9,8,-11,4)(-13,2)(-15,6)(1,9,3,13)(5,11,7,15)
Multiloop annotated with half-edges
10^3_4 annotated with half-edges