On inequalities between unknotting numbers and crossing numbers of spatial embeddings of trivializable graphs and handlebody-knots

TL;DR

This paper explores the relationship between unknotting numbers and crossing numbers in spatial embeddings of certain graphs and handlebody-knots, extending known inequalities and characterizing cases of equality.

Contribution

It extends the inequality relating unknotting and crossing numbers from knots to handlebody-knots and characterizes those handlebody-knots that satisfy the equality.

Findings

Extended the inequality to handlebody-knots.

Characterized handlebody-knots with equality.

Established relations for spatial embeddings of specific graphs.

Abstract

We study relations between unknotting number and crossing number of a spatial embedding of a handcuff-graph and a theta curve. It is well known that for any non-trivial knot $K$ twice the unknotting number of $K$ is less than or equal to the crossing number of $K$ minus one. We show that this is extended to handlebody-knots. We also characterize the handlebody-knots which satisfy the equality.