Unknotting number and connected sums: The knots $4_1$ and $5_1$

Abstract

We show that the knots $K\in \{4_1,5_1\}$ can be paired with a corresponding knot $K^{\prime }$ such that $u(K\#K^{\prime })<u(K)+u(K^{\prime })$. As a consequence unknotting number fails to be additive for these knots. We also provide a candidate knot $K^{\prime }$ for the knot $3_1$.