The conormal torus is a complete knot invariant

TL;DR

This paper demonstrates that the conormal torus, a geometric object associated with knots, uniquely determines the knot up to isotopy or mirror image using microlocal sheaf theory.

Contribution

It establishes that Legendrian isotopy of conormal tori completely characterizes knots, providing a new invariant for knot classification.

Findings

Conormal tori are complete invariants for knots.

Microlocal sheaf theory links Legendrian isotopy to knot isotopy.

Knots with Legendrian isotopic conormal tori are either isotopic or mirror images.

Abstract

We use microlocal sheaf theory to show that if two knots have Legendrian isotopic conormal tori, then the knots are isotopic or mirror images.