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 $M=\mathcal H\cup_{\varphi} \mathcal C(U)$ where $\mathcal H$ is a handlebody, $\mathcal C(U)$ is the mapping cylinder of the orientating two sheeted covering of a non-orientable closed surface $U$ and $\varphi:\partial \mathcal H\to \partial \mathcal C(U)$ 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 $\partial \mathcal H$. Manifolds of this type were extensively studied by J.H. Rubinstein \cite{rubinstein1978one}, where it is shown that any 3-manifold $M$, with a non-vanishing $H_2(M,\frac{\mathbb{Z}}{2\mathbb{Z}})$ will admit such a splitting. Thus the method is quite general. We provide explicit examples of such embeddings in lens spaces $L(2k,q)$ and the trivial circle bundles over orientable closed surfaces, $Σ\times S^1$