How to compute singular homology in integer coefficient of the hypersurface $t^2-1=z^{n}+x(xy-1)$ contained in $A^4$?

I was trying to compute singular homology in integer coefficient of the hypersurface $t^2-1=z^{n}+x(xy-1)$ contained in $A^4$. Can anyone help me computing that? Can anyone tell me some reference where singular homology of some singular affine variety is computed?

How to compute homology of the hypersurface $t^2-1=z^{n}+x(xy-1)$ contained in $A^4$?

How to compute singular homology in integer coefficient of the hypersurface $t^2-1=z^{n}+x(xy-1)$ contained in $A^4$?