The Harary--Hill conjecture, still open after more than 50 years, asserts that the crossing number of the complete graph $K_n$ is $ H(n) = frac 1 4 leftlfloorfrac{mathstrut n}{mathstrut 2}rightrfloor leftlfloorfrac{mathstrut n-1}{mathstrut 2}rightrfloor leftlfloorfrac{mathstrut n-2}{mathstrut 2}rightrfloor leftlfloorfrac{mathstrut n-3}{mathstrut 2}right rfloor$. Abrego et al. introduced the notion of shellability of a drawing $D$ of $K_n$. They proved that if $D$ is $s$-shellable for some $sgeqlfloorfrac{n}{2}rfloor$, then $D$ has at least $H(n)$ crossings. This is the first combinatorial condition on a drawing that guarantees at least $H(n)$ crossings. In this work, we generalize the concept of $s$-shellability to bishellability, where the former implies the latter in the sense that every $s$-shellable drawing is, for any $b leq s-2$, also $b$-bishellable. Our main result is that $(lfloor frac{n}{2} rfloor!-!2)$-bishellability of a drawing $D$ of $K_n$ also guarantees, with a simpler proof than for $s$-shellability, that $D$ has at least $H(n)$ crossings. We exhibit a drawing of $K_{11}$ that has $H(11)$ crossings, is 3-bishellable, and is not $s$-shellable for any $sgeq5$. This shows that we have properly extended the class of drawings for which the Harary-Hill Conjecture is proved. Moreover, we provide an infinite family of drawings of $K_n$ that are $(lfloor frac{n}{2} rfloor!-!2)$-bishellable, but not $s$-shellable for any $sgeqlfloorfrac{n}{2}rfloor$.
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