Given any knot diagam, we can deform it into unknot by switching over arc and under arc of some crossings in the diagram. My question is the following:
Let K be a knot a diagram of a knot, suppose the number of crossings need to exchange in order to obtain an unknot is one, then is it true that the knot diagram K is minimal (i.e K has minimal number of crossings).
On
Suppose $K$ is non-trivial, and let's use $D(K)$ to distinguish $K$ from a particular diagram representing $K$. Let $u(K)$ be the unknotting number of $K$. If $D(K)$ can be unknotted with one crossing change then, $u(K)\leq 1$. If $u(K)=0$ then by definition $K$ is the unknot and so trivial. We've already ruled out this case. It follows that $u(K)=1$ and so $D(K)$ is a minimal unknotting diagram for the knot $K$.
So, this is definitely not true. The unknotting number, which you described, is an invariant of the knot. That means that changing the diagram of the knot you are looking at does not change the value of the invariant. So, adding crossings by some Reidemeister moves will not increase the unknotting number. So, if I take any diagram, lets say of the trefoil, and increase the number of crossings by type I moves, it will still have unknotting number one and not be the minimal diagram.