Knots and non-orientable surfaces in 3-manifolds

TL;DR

The work develops a non-orientable plat framework for knots/links in splittable 3-manifolds by leveraging one-sided Heegaard splittings $M=\mathcal{H} \cup_\varphi \mathcal{C}(U)$ and a non-orientable plat closure of braids in $\partial\mathcal{H}$, enabling a closure representation of any link via the boundary surface braid group. It proves a central theorem: every link in a splittable 3-manifold $M$ is isotopic to a non-orientable plat closure of a braid in $\Sigma_{g-1}$, and provides explicit embeddings and closures in lens spaces $L(2k,q)$ and in $\Sigma_g \times S^1$ using the embedding theory of non-orientable surfaces (e.g., Bredon–Wood). The paper also details concrete constructions in two lens-space families, $L(2k,1)$ and $L(4a{+}4,2a{+}1)$, via tau-curve images that realize minimal-genus non-orientable surfaces and corresponding one-sided splittings, highlighting the method's algebraic and geometric richness. By combining these embeddings with a closure mechanism based on surface braids, the results extend classical plat closures to a broad class of 3-manifolds and illuminate opportunities for further algebraic study using surface braid groups.

Abstract

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, $Σ\times S^1$

Figures (17)

• Figure 1: Mapping cylinder of the Klein bottle.
• Figure 2: Capping curves inside a handlebody.
• Figure 3: Two spheres with connected with a tube.
• Figure 4: Thickening two spheres connected by a tube.
• Figure 5: Thickenings of the cases $g=2$ and $g=3$.

Theorems & Definitions (9)

• Definition 2.1
• Theorem 2.1: rubinstein1978one
• Theorem 2.2: end1992non
• Lemma 3.1
• proof
• Definition 3.1
• Theorem 3.1
• proof
• Theorem 4.1: bredon1969non