I am trying to prove the injective property of group homomorphism between Wirtinger's Presentation of Trefoil Knot and by realizing the trefoil as a torus knot.
I define $ \phi $ such that it takes a to $x_{2}x_{3}x_{2}$ and b to $ x_{2}x_{3}$.
any hint?
It would help to know what you mean by $a,b,x_1,x_2,x_3$. I'm going to guess $a^3=b^2$ for sake of discussion.
Did you already show it was a homomorphism? In this case, it would mean show $\phi(a)^3=\phi(b)^2$.
Did you already show it was surjective? You could do so by showing how to write each $x_i$ in terms of $\phi(a)$ and $\phi(b)$.
For injectivity, you could show $\phi$ has a left inverse $\psi$ (a homomorphism such that $\psi\circ\phi=\mathrm{id})$. Figure out where to send each $x_i$ so that $a=\psi(x_2x_3x_2)$ and $b=\psi(x_2x_3)$, and check the homomorphism property.
One may also proceed by Tietze transformations, of which the homomorphisms referred to above are just an ad hoc approximation. Here is an image to demonstrate the calculation: