Unknotting number and genus of 3-braid knots

TL;DR

This paper investigates the relationship between the unknotting number and genus of 3-braid knots, establishing an inequality and characterizing cases where equality holds.

Contribution

It proves that for 3-braid knots, the unknotting number is at most the genus and characterizes knots where equality occurs.

Findings

u(K) ≤ g(K) for 3-braid knots

Equality u(K) = g(K) characterizes specific knot types

Identifies special classes of knots when equality holds

Abstract

Let $u(K)$ and $g(K)$ denote the unknotting number and the genus of a knot $K$, respectively. For a 3-braid knot $K$, we show that $u(K)\le g(K)$ holds, and that if $u(K)=g(K)$ then $K$ is either a 2-braid knot, a connected sum of two 2-braid knots, the figure-eight knot, a strongly quasipositive knot or its mirror image.