Definition of the cord algebra of knots using Morse thoery

Petrak, Andreas

doi:10.2140/agt.2024.24.2971

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Definition of the cord algebra of knots using Morse thoery

Andreas Petrak

Algebraic & Geometric Topology 24 (2024) 2971–3042

DOI: 10.2140/agt.2024.24.2971

Abstract

The cord algebra of a knot $K$ is isomorphic to the string homology and the Legendrian contact homology of $K$ . The proof of the isomorphism of string homology and cord algebra uses a retraction of broken strings (which are consecutive paths beginning and ending on the knot) on words in linear cords. This suggests a reformulation of the cord algebra using linear cords, which we present. We will define a Morse function such that the binormal linear cords of $K$ are the critical points of degree $0$ , $1$ and $2$ of this function. These critical points give rise to a chain complex of $K$ . Then the cord algebra of $K$ is the degree zero homology of $K$ .

Keywords

knot invariant, Morse theory

Mathematical Subject Classification 2010

Primary: 57M25, 57M27

Secondary: 57R17

References

Publication

Received: 14 July 2019

Revised: 6 November 2022

Accepted: 18 December 2022

Published: 7 October 2024

Authors

Andreas Petrak
German space operation center DLR (Deutsches Zentrum für Luft- und Raumfahrt) Wessling Germany

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Volume 24, issue 6 (2024)