2022/11/19 by Thomas Fiedler, Fiedler, Thomas
Mathematics · #57M25 #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2211.10734
openalex publication_date 2022/11/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Mn 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 Mn 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 Mn. 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.