  The Harary-Hill Conjecture states that for $n\geq 3$ every drawing of $K_n$
has at least \begin{align*}
  H(n) :=
\frac{1}{4}\Big\lfloor\frac{n}{2}\Big\rfloor\Big\lfloor\frac{n-1}{2}\Big\rfloor\Big\lfloor\frac{n-2}{2}\Big\rfloor\Big\lfloor\frac{n-3}{2}\Big\rfloor
\end{align*} crossings. In general the problem remains unsolved, however there
has been some success in proving the conjecture for restricted classes of
drawings. The most recent and most general of these classes is
seq-shellability. In this work, we improve these results and introduce the new
class of semi-pair-shellable drawings. We prove the Harary-Hill Conjecture for
this new class using novel results on $k$-edges. So far, approaches for proving
the Harary-Hill Conjecture for specific classes rely on a fixed reference face.
We successfully apply new techniques in order to loosen this restriction, which
enables us to select different reference faces when considering subdrawings.
Furthermore, we introduce the notion of $k$-deviations as the difference
between an optimal and the actual number of $k$-edges. Using $k$-deviations, we
gain interesting insights into the essence of $k$-edges, and we further relax
the necessity of fixed reference faces.
