The $(4,p)$-arithmetic hyperbolic lattices, $p\ge 2$, in three dimensions

We identify the finitely many arithmetic lattices $\mathrm{\Gamma }$ in the orientation preserving isometry group of hyperbolic $3$-space ${\mathbb{H}}^3$ generated by an element of order $4$ and and element of order $p\ge 2$. Thus $\mathrm{\Gamma }$ has a presentation of the form $\mathrm{\Gamma }\cong ⟨f,g:f^4=g^p=w(f,g)=\cdots =1⟩$ We find that necessarily $p\in \{2,3,4,5,6,\mathrm{\infty }\}$, where $p=\mathrm{\infty }$ denotes that $g$ is a parabolic element, the total degree of the invariant trace field $k\mathrm{\Gamma }=\mathbb{Q}(\{{\text{tr}}^2(h):h\in \mathrm{\Gamma }\})$ is at most $4$, and each orbifold is either a two bridge link of slope $r\mathrm{/}s$ surgered with $(4,0)$, $(p,0)$ Dehn surgery (possibly a two bridge knot if $p=4$) or a Heckoid group with slope $r\mathrm{/}s$ and $w(f,g)=(w_{r\mathrm{/}s}{)}^r$ with $r\in \{1,2,3,4\}$. We give a discrete and faithful representation in $PSL(2,\mathbb{C})$ for each group and identify the associated number theoretic data.