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  1. ; (, Geometry topology)
    Given a knot K in S3, let u−(K) (respectively, u+(K)) denote the minimum number of negative (respectively, positive) crossing changes among all unknotting sequences for K. We use knot Floer homology to construct the invariants l−(K),l+(K) and l(K), which give lower bounds on u−(K),u+(K) and the unknotting number u(K), respectively. The invariant l(K) only vanishes for the unknot, and satisfies l(K) ≥ ν+(K), while the difference l(K) − ν+(K) can be arbitrarily large. We also present several applications towards bounding the unknotting number, the alteration number and the Gordian distance. 
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