ScalingStacks

These definitions are from [LT] and [T].

Proof: The differential $\partial^{0}:\overline{\cal C}^{0}(D)\to\overline{\cal C}^{1}(D)$ is not injective and, hence, $\overline{{\cal H}}^{0}(D)\not=0$ if and only if $D$ is $-$adequate. Similarly for $\overline{{\cal H}}^{n}(D)$ and $+$adequate diagrams (groups $\overline{{\cal H}}^{i}(D)$ were defined at the end of Section 7.1). $\square$

Denote by $c(L)$ the crossing number of $L$. It is the minimal number of crossings in a plane diagram of $L.$

Proof: Let $D$ be a diagram of $L$ with $c(L)$ crossings. Then $\overline{\cal C}^{i}(D)=0$ for $i<0$ and for $i>{\mathrm{hl}}(L).$ Consequently, $\overline{{\cal H}}^{i}(D)=0$ for $i<0$ and $i>{\mathrm{hl}}(L).$

$\square$

Proof: By Proposition 36 $\overline{{\cal H}}^{0}(D)\not=0$ and $\overline{{\cal H}}^{n}(D)\not=0.$ Therefore, $c(L)\geq{\mathrm{hl}}(L)\geq n.$ But since $D$ is an $n$-crossing diagram of $L,$ the crossing number of $L$ is $n.$ $\square$

Corollary 13 was originally obtained by Thistlethwaite (Corollary 3.4 of [T]) through the analysis of the $2$-variable Kauffman polynomial (not to be confused with the Kauffman bracket).