The normalized graded Euler series of HHH recovers HOMFLYPT

Assume the Axiom of Choice, inherited from the comparison HHH is isomorphic to reduced Khovanov-Rozansky homology and the rationality/oriented-link descent of Khovanov-Rozansky homology categorifies the HOMFLYPT polynomial. Let $\sigma$ be a braid word with nonempty braid closure and braid diagram $D$ on $s(D)\ge 1$ strands and closure $\widehat{D}$, and form the Euler characteristic of $HHH$ in the corrected trigrading $(c,h,p)$ with the sign $(-1{)}^c$ on the cohomological (Rouquier) degree and the marked variables of the dictionary $a=-h$, $q=p-h$, $t=c$: $⟨HHH(\sigma )⟩:={\sum }_{h,p,c}(-1{)}^c{t}^{-h}{q}^{p-h}{\dim }_{\mathbb{Q}}HH{H}^{c,h,p}(\sigma )\mathrm{.}$ Then the normalized series $E(\sigma ):=(t{q}^{-1}(1-q^2){)}^{-1}{\sqrt{\alpha }}^{\mathrm{\mid }D{\mathrm{\mid }}_{+}-\mathrm{\mid }D{\mathrm{\mid }}_{-}-s(D)+1}⟨HHH(\sigma )⟩$ is the HOMFLYPT invariant of the closure in the v2 normalization of The normalized Khovanov-Rozansky HOMFLYPT Euler series and Khovanov-Rozansky homology categorifies the HOMFLYPT polynomial: $E(\sigma )=\tilde{F}(\widehat{D})$, the Markov-invariant v2 series with one-strand value $\alpha \mathrm{/}(1-q^{-2})$; here $\mathrm{\mid }D{\mathrm{\mid }}_{+},\mathrm{\mid }D{\mathrm{\mid }}_{-}$ are the crossing numbers of $D$, $s(D)$ its number of strands and $\alpha =-t^{-1}{q}^{-1}$. Equivalently $⟨HHH(\sigma )⟩=t{q}^{-1}(1-q^2){\sqrt{\alpha }}^{s(D)-1-\mathrm{\mid }D{\mathrm{\mid }}_{+}+\mathrm{\mid }D{\mathrm{\mid }}_{-}}\tilde{F}(\widehat{D}),$ so applying the explicit diagram-dependent normalization factor to the HHH Euler series gives the v2 HOMFLYPT series of the closure. That factor is $(t{q}^{-1}(1-q^2){)}^{-1}{\sqrt{\alpha }}^{\mathrm{\mid }D{\mathrm{\mid }}_{+}-\mathrm{\mid }D{\mathrm{\mid }}_{-}-s(D)+1}$; adjoining the trivial factor $\mathbb{Q}[x]$, which multiplies the reduced Euler characteristic by $(1-q^2{)}^{-1}$, recovers the unreduced series of The normalized Khovanov-Rozansky HOMFLYPT Euler series.

Caveats: the formula is stated in the integer grading of the Khovanov-Rozansky theory of The Khovanov-Rozansky complex and trigraded braid homology, in which raw homology changes under stabilization by the specified overall shifts, removed by the displayed series normalization; the sign $(-1{)}^c$ is the one carried by the cohomological (Rouquier) degree, the Hochschild degree entering the weights only through the shifts $k=-h$ and $l=p-h$, and it is read off the global correction of the comparison and verified on the one-strand braid; the HOMFLYPT normalization is the v2 one of the cited items, namely the series $\tilde{F}$ with one-strand value $\alpha \mathrm{/}(1-q^{-2})=t^{-1}q\mathrm{/}(1-q^2)$; the published function $F$ of the categorification theorem uses the distinct normalization $F(\text{unknot})=(t^{-1}-t)\mathrm{/}(q-q^{-1})$. This corollary identifies $HHH$ with the v2 series only and makes no explicit formula comparison between the two Euler normalizations.