  We study Sarkisov links initiated by the toric weighted blowup of a point in
$\mathbb{P}^3$ or $\mathbb{P}^4$ using variation of GIT. We completely classify
which of these initiate Sarkisov links and describe the links explicitly.
Moreover, if $X$ is the toric weighted blowup of $\mathbb{P}^d$ at a point, we
give a simple criterion in terms of the weights of the blowup that
characterises when $X$ is weak Fano.
