Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Constructing embedded surfaces for cellular embeddings of leveled spatial graphs

Senja Barthel, Fabio Buccoliero

TL;DR

The paper introduces leveled spatial graphs and provides a constructive framework to embed such graphs cellularly on a closed orientable surface $\mathcal{S}\subset\mathbb{R}^3$. It proves that leveled graphs with up to four levels admit a cellular embedding via a sphere-and-handles construction, and it develops an algorithmic approach for Hamiltonian leveled graphs (with extensions to non-Hamiltonian and multi-leveled cases) to produce $\mathcal{S}$ when successful. The method includes detailed cylinder-placement rules, intersection-avoidance criteria, and a face-tracing mechanism to verify cellularity, while acknowledging practical limitations and proposing conjectures about broader applicability. The results connect to circular embeddings and open-book framing, offering a pathway to explicit, verifiable realizations of cellular embeddings for a wide class of spatial graphs in 3-space. Overall, the work provides both a theoretical and algorithmic route to realizing cellular embeddings in $\mathbb{R}^3$, with implications for related areas in knot theory and graph embedding conjectures.

Abstract

For a given spatial graph $\mathcal{G} \subset \mathbb{R}^3$, we would like to find a closed orientable surface $\mathcal{S}$ embedded in $\mathbb{R}^3$ in which $\mathcal{G}$ is cellular embedded. However, for general $\mathcal{G}$ this is not possible. We therefore define a property of spatial graphs, called leveled, to show that for leveled spatial graphs with a small number of levels, a surface $\mathcal{S}$ can always be found. The argument is based on decomposing $\mathcal{G}$ into spatial subgraphs that can be placed on a sphere and on cylinders attached as handles, in such a way that the resulting surface contains a cellular embedding of $\mathcal{G}$. We generalize the procedure to an algorithm that, if successful, constructs $\mathcal{S}$ for leveled spatial graphs with any number of levels. We conjecture that all connected leveled embeddings can be cellular embedded with the presented algorithm.

Constructing embedded surfaces for cellular embeddings of leveled spatial graphs

TL;DR

The paper introduces leveled spatial graphs and provides a constructive framework to embed such graphs cellularly on a closed orientable surface . It proves that leveled graphs with up to four levels admit a cellular embedding via a sphere-and-handles construction, and it develops an algorithmic approach for Hamiltonian leveled graphs (with extensions to non-Hamiltonian and multi-leveled cases) to produce when successful. The method includes detailed cylinder-placement rules, intersection-avoidance criteria, and a face-tracing mechanism to verify cellularity, while acknowledging practical limitations and proposing conjectures about broader applicability. The results connect to circular embeddings and open-book framing, offering a pathway to explicit, verifiable realizations of cellular embeddings for a wide class of spatial graphs in 3-space. Overall, the work provides both a theoretical and algorithmic route to realizing cellular embeddings in , with implications for related areas in knot theory and graph embedding conjectures.

Abstract

For a given spatial graph , we would like to find a closed orientable surface embedded in in which is cellular embedded. However, for general this is not possible. We therefore define a property of spatial graphs, called leveled, to show that for leveled spatial graphs with a small number of levels, a surface can always be found. The argument is based on decomposing into spatial subgraphs that can be placed on a sphere and on cylinders attached as handles, in such a way that the resulting surface contains a cellular embedding of . We generalize the procedure to an algorithm that, if successful, constructs for leveled spatial graphs with any number of levels. We conjecture that all connected leveled embeddings can be cellular embedded with the presented algorithm.
Paper Structure (13 sections, 17 theorems, 2 equations, 16 figures, 2 algorithms)

This paper contains 13 sections, 17 theorems, 2 equations, 16 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Cellular embeddings of leveled spatial graphs with few levels
  4. Constructing the surface
  5. Placing cylinders under fragments
  6. Attaching cylinders to the surface
  7. Conditions to avoid self-intersections of the surface
  8. Conditions to ensure cellularity
  9. The algorithm for the surface construction
  10. Results of the algorithm
  11. Outlook
  12. Acknowledgement
  13. Bibliography

Key Result

Lemma 1

Let $\mathcal{G}$ be a spatial graph with a vertex $v$ that is the endpoint of two fragments $f_1, f_2$ with respect to a cycle $C$. It is always possible to split a vertex $v$ into two vertices $v_1$ and $v_2$ by adding an edge to $C$ such that $v_1$ is an endpoint of $f_1$ and $v_2$ is an endpoint

Figures (16)

  • Figure 1: A hamiltonian leveled spatial graph with spine list $(1^1, 2^2, 3^1, 3^2, 2^1, 3^2, 3^1, 1^2, 2^1, 2^2, 2^3, 1^2, 2^3, 1^1)$. The superscripts distinguish fragments at same levels.
  • Figure 2: A fragment $f$ with a right halftwist with respect to the cylinder $P$.
  • Figure 3: The endpoints of the fragments can be moved along the connected components of the boundary of the cylinder $P$ which is not attached to the surface.
  • Figure 4: An upper cylinder $P$ with a fragment $f$ with upper endpoints and a lower cylinder $P'$ with a fragment $g$ with upper endpoints.
  • Figure 5: The situation described in cases 1 and 2 of \ref{['lem: levels of fragments on cylinder']}.
  • ...and 11 more figures

Theorems & Definitions (49)

  • Definition 1
  • Remark 1
  • Lemma 1
  • proof
  • Proposition 1
  • proof
  • Corollary 1
  • Example 1
  • Definition 2
  • Definition 3
  • ...and 39 more