Unknotting number and genus of 3-braid knots
Eon-Kyung Lee, Sang-Jin Lee

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 and denote the unknotting number and the genus of a knot , respectively. For a 3-braid knot , we show that holds, and that if then 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.
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