End Sn.
Section FG.
(* We tie it together *)
Fixpoint fg (n : nat) {struct n} : ((K1.knot -> K2.knot) * (K2.knot -> K1.knot)) :=
match n with
| 0 => (f_Z, g_Z)
| S n => match fg n with (fn, gn) => (f_Sn gn, g_Sn fn) end
end.

Lemma fg_fg_eq: forall n k2,
level k2 < n ->
snd (fg (level k2)) k2 = snd (fg n) k2.