Consider the moves into and out of a1, one must move from one of b3 or c2 and the other must move to the one of b3 and c2 not moved out of to reach a1. That is the two moves through a1 are (b3-a1, a1-c2) or (c2-a1, a1-c3)
But a similar argument applies to square d4 as to a1, so any closed path of knights moves that passes through a1 and d4 and does not visit any square more than once cannot visit any of the squares a2, a3, a4, b1, b2, b4, c1. c3, c4, d1, d2, d3. Hence there is no closed knights tour on the 4x4 chess board that visits each square once only.