Polynomial invariants which can distinguish the orientations of knots

TL;DR

This paper introduces new polynomial invariants capable of distinguishing the orientations of classical knots without relying on the knot group, utilizing geometric features like meridian and longitude.

Contribution

It presents the first knot polynomials that differentiate orientations using geometric data and finite type invariants, constructed via combinatorial 1-cocycles on moduli spaces.

Findings

Polynomial invariants distinguish knot orientations.

Construction of integer-valued 1-cocycles using Gauss diagrams.

Evaluation of invariants on canonical loops yields new polynomial invariants.

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 $T$.