New perspectives on self-linking

TL;DR

This paper introduces a new homotopy-theoretic approach to classical knots via evaluation maps on configuration spaces, leading to a novel geometric interpretation of the second coefficient of the Conway polynomial and applications to knot complexity.

Contribution

It develops the homotopy classification of knots through evaluation maps, computes the associated models, and links a new invariant to finite-type invariants and geometric properties.

Findings

The third mapping space model yields an integer-valued invariant.

This invariant matches the second coefficient of the Conway polynomial.

New applications relate the invariant to quadrisecants and knot complexity.

Abstract

We initiate the study of classical knots through the homotopy class of the n-th evaluation map of the knot, which is the induced map on the compactified n-point configuration space. Sending a knot to its n-th evaluation map realizes the space of knots as a subspace of what we call the n-th mapping space model for knots. We compute the homotopy types of the first three mapping space models, showing that the third model gives rise to an integer-valued invariant. We realize this invariant in two ways, in terms of collinearities of three or four points on the knot, and give some explicit computations. We show this invariant coincides with the second coefficient of the Conway polynomial, thus giving a new geometric definition of the simplest finite-type invariant. Finally, using this geometric definition, we give some new applications of this invariant relating to quadrisecants in the knot and to complexity of polygonal and polynomial realizations of a knot.