Does the factorization operation for composite knots applicable for any projection of the knot?

Question

Let $K$ be a knot in 3-space. We know that if $K$ is not prime (composite) then $K$ can be represented as a connected sum of two knots, both are non-trivial. My question is : does this apply to any arbitrary projection of $K$ in a plane? In other words, if I take any projection of $K$, then I can decompose it into two non-trivial knots by the decomposition operation or in some projections, we have to apply some Reidemeister moves first before applying the decomposition operation.

Answer 1

I will make an assumption about what you mean by "decompose it into two non-trivial knots by the decomposition algorithm." Let's call a diagram composite if there is a simple closed curve in the plane intersecting the diagram exactly twice away from the crossings such that both the interior and exterior of that simple closed curve contain crossings of the diagram. I will interpret your question as follows:

Rephrased question: Are all diagrams of composite knots composite?

If I've interpreted your question correctly, then the answer is no. Here is a counterexample of the connected sum of two trefoils.