Turaev cobracket, synthetic geometry and applications
Abstract
A teardrop geodesic on a hyperbolic surface is a closed curve that is geodesic except at exactly one point, called its vertex. One imagines a closed curve with one point held down as it deforms to a geodesic. For each teardrop geodesic, we give an explicit geometric construction of the closed geodesic in its free homotopy class. This construction also yields a relation between the length of the teardrop geodesic, the angle at its vertex, and the length of its geodesic representative. We then apply the construction to a self-intersecting closed geodesic at a chosen self-intersection point. Since, at that point, the geodesic splits into two directed teardrop geodesics, our construction gives a geometric description of Turaev's cobracket in terms of geodesic representatives. Using this description we prove that if two self-intersection points yield a pair of canceling terms in the cobracket, then the angle at these points and the lengths of the small loops determined are the same.
Related
- Synthetic geometry of differential equations: I. Jets and comonad structure
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- Chromatic Spherical and Renormalized Reshetikhin-Turaev Invariants for Drinfeld Centers
BibTeX
@misc{chas2026turaev,
title = {{Turaev cobracket, synthetic geometry and applications}},
author = {Moira Chas and Arpan Kabiraj},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.17235},
eprint = {2610.17235},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.17235},
}
RIS
TY - UNPB
TI - Turaev cobracket, synthetic geometry and applications
AU - Moira Chas
AU - Arpan Kabiraj
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.17235
UR - https://arxiv.org/abs/2610.17235
AB - A teardrop geodesic on a hyperbolic surface is a closed curve that is geodesic except at exactly one point, called its vertex. One imagines a closed curve with one point held down as it deforms to a geodesic. For each teardrop geodesic, we give an explicit geometric construction of the closed geodesic in its free homotopy class. This construction also yields a relation between the length of the teardrop geodesic, the angle at its vertex, and the length of its geodesic representative. We then apply the construction to a self-intersecting closed geodesic at a chosen self-intersection point. Since, at that point, the geodesic splits into two directed teardrop geodesics, our construction gives a geometric description of Turaev's cobracket in terms of geodesic representatives. Using this description we prove that if two self-intersection points yield a pair of canceling terms in the cobracket, then the angle at these points and the lengths of the small loops determined are the same.
ER -