ScalingStacks

We specialize now to the case of the classical Artin braid groups considered by Khovanov in [Kh]. Note that Khovanov conjectured ten years ago that the $F_{b}$’s should give rise to interesting link invariants.

We take here $V=(\bigoplus_{i=1}^{n}ke_{i})/k(e_{1}+\cdots e_{n})$, the reflection representation of $W={\mathfrak{S}}_{n}$, with $S=\{(1,2),\ldots,(n-1,n)\}$. Let $P_{n}=k[V]=k[\alpha_{1},\ldots,\alpha_{n-1}]$, where $\alpha_{i}=X_{i+1}-X_{i}$. We put $B_{n}=B_{(W,S)}$.

Let $P_{\infty}=\lim_{n}P_{n}$, where the limit is taken over the morphisms of $P_{n}$-algebras $\rho_{n}:P_{n+1}\to P_{n},\ \alpha_{n}\mapsto 0$. This provides functors between derived categories

Passing to homology, we recover the following result of Khovanov [Kh]. Khovanov identifies the invariant as the Khovanov-Rozansky homology, a categorification of the HOMFLYPT polynomial.

Note that $X_{1}=1$, where $1\in B_{1}$ corresponds to the trivial knot.

We define now a two variables invariant $Y_{b}=(X_{b})_{|t_{3}^{1/2}=\sqrt{-1}}$. The following corollary shows that $Y_{b}$ is the HOMFLYPT polynomial, as expected.