### Instruction: What is the minimum number of crossings required for a non-trivial knot with 6 edges?

Hint: Use the Knot Theory concept that relates the number of edges, vertices, and faces to the minimum number of crossings.### Response: In knot theory, we can use the Euler's formula for planar graphs to relate the number of edges (E), vertices (V), and faces (F) in a knot diagram. The formula is:

V - E + F = 2

For a non-trivial knot with 6 edges, we can use this formula to find the minimum number of crossings. In a knot diagram, each crossing is represented by a vertex, and each edge connects two vertices. Since there are 6 edges, we have:

2V = 2E = 12

So, V = 6. Now, we can plug this value into Euler's formula:

6 - 6 + F = 2

F = 2

Since there are 2 faces, we can now determine the minimum number of crossings. In a non-trivial knot, each face must have at least 4 edges. Therefore, the minimum number of crossings (C) can be found by dividing the total number of edges by 2:

C = E/2 = 6/2 = 3

So, the minimum number of crossings required for a non-trivial knot with 6 edges is 3.