Unknotting numbers and crossing numbers of spatial embeddings of a planar graph

TL;DR

This paper investigates the relationship between unknotting number and crossing number in spatial embeddings of planar graphs, revealing cases where the unknotting number exceeds half the crossing number, contrary to known inequalities.

Contribution

It demonstrates the existence of planar graph embeddings where the unknotting number surpasses half the crossing number, challenging previous assumptions.

Findings

Existence of planar graph embeddings with unknotting number > half the crossing number

Analysis of relations between unknotting and crossing numbers in spatial graph embeddings

Counterexamples to the known inequality for links

Abstract

It is known that the unknotting number $u(L)$ of a link $L$ is less than or equal to half the crossing number $c(L)$ of $L$. We show that there are a planar graph $G$ and its spatial embedding $f$ such that the unknotting number $u(f)$ of $f$ is greater than half the crossing number $c(f)$ of $f$. We study relations between unknotting number and crossing number of spatial embedding of a planar graph in general.