PreprintCC BY 4.0arXiv
On the A∞-extension of Bar-Natan multicurves
Abstract
Kotelskiy, Watson and Zibrowius show that the Bar-Natan invariant unicodex0414(T) associated to a $4$-ended oriented tangle T extends to a twisted complex unicodex0414∞(T) over an A∞-enhancement of the Bar-Natan algebra. We prove that unicodex0414∞(T) itself is a tangle invariant, settling a conjecture from their work. We deduce this from an invariance statement for a \emph{matrix multifactorization} ℳ(T), generalizing a construction of Ballinger to the coefficient ring ℤ[G]. This multifactorization is well defined up to a notion of 1-homotopy equivalence. The proof rests on explicit formulas for special deformation retracts of Koszul matrix factorizations, which realize delooping in this setting and may be of independent interest.
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BibTeX
@misc{mejiagomez2026aextension,
title = {{On the $A_\infty$-extension of Bar-Natan multicurves}},
author = {Tomás Mejía-Gómez},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10262},
eprint = {2610.10262},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10262},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On the A_∞-extension of Bar-Natan multicurves
AU - Tomás Mejía-Gómez
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.10262
UR - https://arxiv.org/abs/2610.10262
AB - Kotelskiy, Watson and Zibrowius show that the Bar-Natan invariant unicodex0414(T) associated to a $4$-ended oriented tangle T extends to a twisted complex unicodex0414^∞(T) over an A_∞-enhancement of the Bar-Natan algebra. We prove that unicodex0414^∞(T) itself is a tangle invariant, settling a conjecture from their work. We deduce this from an invariance statement for a \emph{matrix multifactorization} ℳ(T), generalizing a construction of Ballinger to the coefficient ring ℤ[G]. This multifactorization is well defined up to a notion of 1-homotopy equivalence. The proof rests on explicit formulas for special deformation retracts of Koszul matrix factorizations, which realize delooping in this setting and may be of independent interest.
ER -