Abstract

We use microlocal sheaf theory to show that knots can only have Legendrian isotopic conormal tori if they themselves are isotopic or mirror images.

Highlights

  • A knot K ⊂ R3 determines, by taking the unit conormals, a torus in the unit cosphere bundle

  • The smooth type of this embedding TK ⊂ S∗R3 knows nothing about the knot. This embedding is that of a Legendrian in a contact manifold, and a smooth isotopy of knots K ∼ K induces a Legendrian isotopy of conormals TK ∼ TK

  • The Legendrian isotopy type of TK is a topological invariant of K

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Summary

Introduction

A knot K ⊂ R3 determines, by taking the unit conormals, a torus in the unit cosphere bundle. One can study the geometry of Legendrians in cosphere bundles by assigning, to a Legendrian Λ ⊂ S∗ M, the category of sheaves with microsupport in Λ This approach reaches back in some sense to the microlocal analysis of Sato and Hormander, is technically built on the microlocal sheaf theory of Kashiwara and Schapira [KS], and was applied to study symplectic geometry by Tamarkin [Tam] and subsequent authors [GKS, Gui, Gui, Gui, Chi, STZ, STWZ, STW]. The sheaf quantization of this contact isotopy [GKS] gives an equivalence of derived categories shTK (R3) ∼= shTK (R3) These categories of sheaves carry information about the fundamental group of the knot complement, which (when marked by the peripheral subgroup) determines the knot [Wal]. Other comparisons can be extracted from [Cor, Cor2], and [BEY]

Some review of sheaf theory
Recovering the fundamental group
Recollections on μHom
Longitudes and meridians
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