Multi-variable Alexander polynomial and solvable Lie algebras

Arkady Vaintrob
Department of Mathematics, New Mexico State University

ABSTRACT


It is known that the Alexander-Conway knot polynomial is the Kontsevich integral applied to weight systems coming from the Lie superalgebra $gl(1|1)$. We show that the corresponding link invariant is related to the multi- variable Alexander polynomial $\Delta$. In particular, it implies that, like many other link invariants, $\Delta$ can be expressed by means of Vassiliev invariants.

Moreover, we show that Vassiliev invariants coming from any solvable Lie algebra (or Lie superalgebra) belong to the subalgebra generated this implies that the space of Vassiliev invariants coming from Lie superalgebras is strictly larger than from all Lie algebras with invariant inner product.