The normalized Khovanov-Rozansky HOMFLYPT Euler series

The exponent is well defined. The numbers $\mathrm{\mid }D{\mathrm{\mid }}_{+}$ and $\mathrm{\mid }D{\mathrm{\mid }}_{-}$ are determined by the braid word of $D$ by [F2] and depend only on the diagram, and $s(D)=n$ for $D\subset B_n$; hence the integer $\mathrm{\mid }D{\mathrm{\mid }}_{+}-\mathrm{\mid }D{\mathrm{\mid }}_{-}-s(D)+1$ is a well-defined function of the braid diagram. The power ${\sqrt{\alpha }}^m$ is defined for every $m\in \mathbb{Z}$ because $\alpha$ is invertible in $\mathbb{Z}[q^{\pm 1},t^{\pm 1}]$ and $\sqrt{\alpha }$ and ${\alpha }^{-1\mathrm{/}2}$ are units of $T_0$ by construction. The remaining issue is that the Euler series is in this localized ring, proved next.

The two stabilizations move the exponent controllably. For the positive stabilization $D{\sigma }_n$ of a diagram on $n$ strands one has $\mathrm{\mid }D{\sigma }_n{\mathrm{\mid }}_{+}=\mathrm{\mid }D{\mathrm{\mid }}_{+}+1$ and $s=n+1$, so the exponent $\mathrm{\mid }D{\mathrm{\mid }}_{+}-\mathrm{\mid }D{\mathrm{\mid }}_{-}-s+1$ is unchanged; for the negative stabilization $D{\sigma }_n^{-1}$ one has $\mathrm{\mid }D{\sigma }_n^{-1}{\mathrm{\mid }}_{-}=\mathrm{\mid }D{\mathrm{\mid }}_{-}+1$ and the exponent decreases by $2$. This is the arithmetic reason for the choice of normalization, and it is used in the categorification theorem together with the stabilization values of the Euler series, not asserted as invariance here.

The rational coefficient ring. For a nonempty braid closure, the finitely many resolution complexes in [F3] are finite complexes of finite free modules over $\mathbb{Q}[x_1,\dots ,x_N]$, after the universal $(a,0)$ row is removed. Only finitely many first internal degrees occur. This polynomial ring is Noetherian by [F4], since $\mathbb{Q}$ has only the ideals $0,\mathbb{Q}$; thus their kernels, images and cohomology, and then the cohomology of the finite crossing cube, are finitely generated in each first internal degree. Split each second internal grading into its two parity classes and set $z=q^2$. Regrading a parity class to integer degrees makes it a finite standard graded polynomial module. Hilbert-Serre [F4], under AC, writes each such series as a Laurent polynomial in $q$ divided by a power of $1-q^2$. There are finitely many cochain and first internal degrees, so their signed $t$-weighted sum is in $\mathbb{Z}[t^{\pm 1},q^{\pm 1},(1-q^2{)}^{-1}]$. Multiplication by the specified root power yields $\tilde{F}(D)\in T_0$. The rational expressions are expanded using $(1-q^2{)}^{-1}={\sum }_{d\ge 0}{q}^{2d}$ when read as formal series. The zero-strand empty diagram is excluded from this normalized-link definition: its raw complex is $\mathbb{Q}[a]$, with $\mathrm{deg}a=(2,0)$, and its raw v2 Euler is $(1-t^2{)}^{-1}$. Substituting $s=e=0$ into the normalization expression would formally give $\sqrt{\alpha }\mathrm{/}(1-t^2)$ in the larger localization adjoining $(1-t^2{)}^{-1}$; this is not identified with an empty-link HOMFLYPT value.

The unknot value. For the one-strand diagram $D$ of the unknot one has $\mathrm{\mid }D{\mathrm{\mid }}_{+}=\mathrm{\mid }D{\mathrm{\mid }}_{-}=0$ and $s(D)=1$, so the exponent is $0$ and $\tilde{F}(D)=⟨D⟩$. The direct computation of the one-mark circle gives $H(D)\cong \mathbb{Q}[x]\{-1,1\}$ and $⟨D⟩={\sum }_{m\ge 0}{t}^{-1}{q}^{1+2m}=t^{-1}q\mathrm{/}(1-q^2)=t^{-1}\mathrm{/}(q^{-1}-q)$; with $\alpha =-t^{-1}{q}^{-1}$ one has $\alpha \mathrm{/}(1-q^{-2})=-t^{-1}{q}^{-1}{q}^2\mathrm{/}(q^2-1)=t^{-1}q\mathrm{/}(1-q^2)$, so $\tilde{F}(D)=\alpha \mathrm{/}(1-q^{-2})$, the normalization recorded in section 7 of the source.