Unknotting numbers for prime $θ$-curves up to seven crossings

Dorothy Buck, Danielle O'Donnol

Published 2017-10-14Version 1

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, excepting $\mathbf{7_5,7_{22}, 7_{24}},$ and $\mathbf{7_{58}}$ which have unknotting number $1$ or $2$. 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.