Knots and non-orientable surfaces in 3-manifolds

In this article, we propose a new approach for describing and understanding knots and links in a 3-manifold through the use of an embedded non-orientable surface. Specifically, we define a plat-like representation based on this non-orientable surface. The method applies to manifolds of the form M = H ∪ φ C ( U ) M=\mathcal H\cup _{\varphi } \mathcal C(U) , where H \mathcal H is a handlebody, C ( U ) \mathcal C(U) is the mapping cylinder of the orientable two sheeted covering of a non-orientable closed surface U U and φ : ∂ H → ∂ C ( U ) \varphi :\partial \mathcal H\to \partial \mathcal C(U) is an attaching homeomorphism. We show that, by fixing such a splitting any link in the manifold can be represented as a plat-like closure of an element of the surface braid group of ∂ H \partial \mathcal H . The method is quite general, since any 3-manifold M M , with a nonvanishing H 2 ( M , Z 2 Z ) H_2(M,\frac {\mathbb {Z}}{2\mathbb {Z}}) admits such a splitting. We provide explicit examples of such a splitting in lens spaces L (