The definition of mirror image of a knot $K$ is as follows:
By the mirror image of a knot $K$ we mean the image of $K$ under the reflection $\mathscr{R}$ defined by $(x, y, z) \rightarrow (x, y, -z).$
But I do not understand why its inverse is itself, could anyone explain this for me please?
We have $(K \circ K)(x,y,z)=K(x,y,-z)=(x,y,-(-z))=(x,y,z).$
Hence $K^{-1}=K.$