Polynomial invariants which can distinguish the orientations of knots

Abstract

This paper contains the first knot polynomials which can distinguish the orientations of classical knots and which make no excplicit use of the knot group. But they make extensive use of the meridian and of the longitude in a geometric way. Let $M$ be the topological moduli space of long knots up to regular isotopy, and for any natural number $n > 1$ let $M_n$ be the moduli space of all n-cables $nK$ of framed long knots $K$ which are twisted by a given string link $T$ to close to a knot in the solid torus, with a marked point on the knot at infinity. First we construct integer valued combinatorial 1-cocycles for $M_n$ by using Gauss diagram formulas for finite typ invariants. We observe then that our 1-cocycles allow to fix certain crossings of $nK$ as local parameters of the 1-cocycles. Finally, we transform the local parameter into an unordered set of global parameters by following the crossings in the isotopy. We evaluate now the 1-cocycles on a canonical loop in $M_n$. The outcome are polynomial valued invariants of $K$, where the variables are indexed by finite type invariants and by regular isotopy types of string links $T$.