Spatial graph as connected sum of a planar graph and a braid

TL;DR

This paper demonstrates that any finite spatial graph can be decomposed into a connected sum of a planar graph and a braid, providing new insights into their fundamental group structure in three-dimensional space.

Contribution

It introduces a novel decomposition of finite spatial graphs into a planar graph and a braid, facilitating the analysis of their fundamental groups.

Findings

Any finite spatial graph is a connected sum of a planar graph and a braid.

This decomposition allows for easier computation of the fundamental group of the graph's complement.

The paper provides a method to find generators and relations for these fundamental groups.

Abstract

In this paper we show that every finite spatial graph is a connected sum of a planar graph, which is a forest, i.e. disjoint union of finite number of trees and a tangle. As a consequence we get that any finite spatial graph is a connected sum of a planar graph and a braid. Using these decompositions it is not difficult to find a set of generators and defining relations for the fundamental group of compliment of a spatial graph in 3-space $\mathbb{R}^3$.