Galois action on knots I: Action of the absolute Galois group

TL;DR

This paper introduces profinite knots to explore the arithmetic structures of knots and establishes an action of the absolute Galois group on these profinite knots, extending the Grothendieck-Teichmüller group's action.

Contribution

It defines profinite knots and demonstrates a rigorous Galois action on them, extending known group actions from braid groups to knots.

Findings

Profinite knots extend classical knots with new algebraic properties.

Established a Galois group action on profinite knots.

Extended the Grothendieck-Teichmüller group's action from braid groups to knots.

Abstract

Our aim of this and subsequent papers is to enlighten (a part of, presumably) arithmetic structures of knots. This paper introduces a notion of profinite knots which extends topological knots and shows its various basic properties. Particularly an action of the absolute Galois group of the rational number field on profinite knots is rigorously established, which is attained by our extending the action of Drinfeld's Grothendieck-Teichm\"{u}ller group on profinite braid groups into on profinite knots.