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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
Petrak, Andreas 1
@article{10_2140_agt_2024_24_2971,
author = {Petrak, Andreas},
title = {Definition of the cord algebra of knots using {Morse} thoery},
journal = {Algebraic and Geometric Topology},
pages = {2971--3042},
publisher = {mathdoc},
volume = {24},
number = {6},
year = {2024},
doi = {10.2140/agt.2024.24.2971},
url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2024.24.2971/}
}
TY - JOUR AU - Petrak, Andreas TI - Definition of the cord algebra of knots using Morse thoery JO - Algebraic and Geometric Topology PY - 2024 SP - 2971 EP - 3042 VL - 24 IS - 6 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.2140/agt.2024.24.2971/ DO - 10.2140/agt.2024.24.2971 ID - 10_2140_agt_2024_24_2971 ER -
Petrak, Andreas. Definition of the cord algebra of knots using Morse thoery. Algebraic and Geometric Topology, Tome 24 (2024) no. 6, pp. 2971-3042. doi: 10.2140/agt.2024.24.2971
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