Unknotting numbers for prime $\theta$-curves up to seven crossings

TL;DR

This paper determines the exact unknotting numbers for prime theta-curves up to seven crossings, introduces new methods for obstructing unknotting number 1, and explores the subadditivity of unknotting numbers in spatial graphs.

Contribution

It provides the first complete determination of unknotting numbers for prime theta-curves up to seven crossings and introduces novel obstruction techniques.

Findings

Exact unknotting numbers for all theta-curves in the Litherland-Moriuchi Table.

Unknotting crossing changes are demonstrated for all these curves.

New methods for obstructing unknotting number 1 in theta-curves are introduced.

Abstract

Determining unknotting numbers is a large and widely studied problem. We consider the more general question of the unknotting number of a spatial graph. We show the unknotting number of spatial graphs is subadditive. Let $g$ be an embedding of a planar graph $G$, then we show $u(g) \geq \max\{u(s) |$ $s$ is a non-overlapping set of constituents of $g\}$. Focusing on $\theta$-curves, we determine the exact unknotting numbers of the $\theta$-curves in the Litherland-Moriuchi Table. Additionally, we demonstrate unknotting crossing changes for all of the curves. In doing this we introduce new methods for obstructing unknotting number $1$ in $\theta$-curves.