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The figure-eight knot is the unique knot of crossing number of a knot four. It is prime, alternating, and has Alexander polynomial of a knot $-t+3-t^{-1}$ up to a unit.

Take the twist knot whose standard diagram has $2020$ half-twists in its twist region and two crossings in its clasp. Every nontrivial twist knot has Seifert genus one. This diagram is a reduced alternating knot diagram, so the Tait crossing-number theorem says that its $2020+2=2022$ crossings realize the crossing number of a knot. Thus this knot has $g_s(K_2)=1$ and $c(K_2)=2022$.

The trefoil is the torus knot $T_{2,3}$. It has Seifert genus one, crossing number of a knot three, and Alexander polynomial of a knot $t-1+t^{-1}$ up to a unit.