  At the HL-LHC and future high energy colliders, a sample of a billion top
quarks will be produced, allowing precision searches for new physics in top
quark decay and production. To aid in this endeavor, we characterize the
independent three and four point on-shell amplitudes involving top quarks,
under the assumption of $SU(3)_c\times U(1)_{em}$ invariance. The four point
amplitudes factorize into primary and descendent amplitudes, where descendants
are primaries multiplied by Mandelstam variables. By enumerating the allowed
amplitudes, we can check for amplitude redundancies to find the number of
independent terms and convert those into a Lagrangian which parameterizes these
amplitudes. These results are then cross checked by utilizing the Hilbert
series to count the number of independent Lagrangian operators. Interestingly,
we find situations where the Hilbert series has cancellations which, if
na\"{i}vely interpreted, would lead to the incorrect conclusion that that there
are no primary operators at a given mass dimension. We characterize the four
fermion ($ffff$) and two fermion, two gauge boson ($ffVV$) operators
respectively up to dimension 12 and 13. Finally, by combining unitarity bounds
on the coupling strengths and simple estimates of the branching ratio
sensitivities, we highlight interesting amplitudes for top quark decay that
should be studied more closely at the HL-LHC. Of those highlighted, there are
both new charge current and flavor changing neutral current decays that occur
at dimension 8 and 10 in SMEFT.
