Chern-Simons factorization algebras and knot polynomials

This work identifies the Reshetikhin-Turaev invariant of links in terms of a trace map on factorization homology. In particular, to recover the knot invariants associated to Chern-Simons theories, we construct a filtered ${\mathcal{E}}_3$-algebra ${\mathcal{A}}^{\lambda }$ by BV quantization of Chern-Simons theory for a semi-simple Lie algebra $\mathfrak{g}$ with invariant pairing~$\lambda$, and we prove that a finite-dimensional representation $V$ of the Drinfeld-Jimbo quantum group $U_{\mathrm{\hslash }}\mathfrak{g}$ defines a perfect ${\mathcal{A}}^{\lambda }$ module~$\mathcal{V}$. For any framed link $K$ in ${\mathbb{R}}^3$, we then prove that there is an equality $${\int }_{K\subset {\mathbb{R}}^3}\mathrm{t}\mathrm{r}(V)=Z_V(K\subset {\mathbb{R}}^3)$$ between the factorization homology trace for $V$ and the Reshetikhin-Turaev link invariant determined by~$V$.