Given a knot , let (respectively, ) denote the minimum number of negative (respectively, positive) crossing changes among all unknotting sequences for . We use knot Floer homology to construct the invariants , and , which give lower bounds on , and the unknotting number , respectively. The invariant only vanishes for the unknot, and satisfies , while the difference can be arbitrarily large. We also present several applications towards bounding the unknotting number, the alteration number and the Gordian distance.
"Knot Floer homology and the unknotting number." Geom. Topol. 24 (5) 2435 - 2469, 2020. https://doi.org/10.2140/gt.2020.24.2435