Transverse knot explained

In mathematics, a transverse knot is a smooth embedding of a circle into a three-dimensional contact manifold such that the tangent vector at every point of the knot is transverse to the contact plane at that point.

Any Legendrian knot can be C⁰-perturbed in a direction transverse to the contact planes to obtain a transverse knot. This yields a bijection between the set of isomorphism classes of transverse knots and the set of isomorphism classes of Legendrian knots modulo negative Legendrian stabilization.

References

• Book: Geiges , Hansjörg . An introduction to contact topology; Volume 109 of Cambridge studies in advanced mathematics. Hansjörg Geiges . 2008 . Cambridge University Press . 978-0-521-86585-2 . 94 .
• J. Epstein, D. Fuchs, and M. Meyer, Chekanov–Eliashberg invariants and transverse approximations of Legendrian knots, Pacific J. Math. 201 (2001), no. 1, 89–106.