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

TL;DR

This paper demonstrates that the unknotting number is not additive for certain knots, specifically $4_1$ and $5_1$, by constructing pairs with knots $K^ extprime$ that reduce the sum of their individual unknotting numbers.

Contribution

It provides explicit examples showing the failure of unknotting number additivity for specific knots and suggests a candidate for the unknotting number of the trefoil knot.

Findings

Unknotting number is not additive for knots $4_1$ and $5_1$.

Constructed pairs of knots with reduced combined unknotting number.

Proposed a candidate knot for the unknotting number of the trefoil knot.

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 $3_1$.