Surface Braid Invariants from Yang-Baxter Operators and Their Cohomology
Abstract
Surface braids are a generalization of braids to surfaces in the 4-disk. They are described by planar graphs called charts, and sequences of braid words called braid systems, that represent monodromies of branch points in projections. Yang-Baxter operators (YBOs) have been used extensively in knot theory for constructions of Jones and other polynomials, defined by braid group representations. We propose to use YBOs for constructing surface braid invariants through braid systems. The intersections of eigenspaces are used for this construction. We further use 3-cocycles of the Yang-Baxter cohomology theories via charts, in an analogue of quandle cocycle invariants but more generally utilizing tensor products of modules. An approach towards defining knotted surface invariants through closures of surface braids is discussed.
Related
- The Classification of the Rota-Baxter Operators on Four-dimensional Grassmann Algebra
- NS-Leibniz bialgebras, the Yang--Baxter equation and relative Rota--Baxter operators
- Extremal Rota-Baxter operators on matrix algebras
- Rota---Baxter operators on extraspecial groups
- Restricted (Relative) Rota-Baxter operators on restricted Lie algebras and restricted Lie triple systems and related structures
BibTeX
@misc{saito2026surface,
title = {{Surface Braid Invariants from Yang-Baxter Operators and Their Cohomology}},
author = {Masahico Saito and Emanuele Zappala},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08837},
eprint = {2610.08837},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08837},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Surface Braid Invariants from Yang-Baxter Operators and Their Cohomology
AU - Masahico Saito
AU - Emanuele Zappala
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.08837
UR - https://arxiv.org/abs/2610.08837
AB - Surface braids are a generalization of braids to surfaces in the 4-disk. They are described by planar graphs called charts, and sequences of braid words called braid systems, that represent monodromies of branch points in projections. Yang-Baxter operators (YBOs) have been used extensively in knot theory for constructions of Jones and other polynomials, defined by braid group representations. We propose to use YBOs for constructing surface braid invariants through braid systems. The intersections of eigenspaces are used for this construction. We further use 3-cocycles of the Yang-Baxter cohomology theories via charts, in an analogue of quandle cocycle invariants but more generally utilizing tensor products of modules. An approach towards defining knotted surface invariants through closures of surface braids is discussed.
ER -