#### Volume 24, issue 5 (2020)

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Knot Floer homology and the unknotting number

### Akram Alishahi and Eaman Eftekhary

Geometry & Topology 24 (2020) 2435–2469
##### Abstract

Given a knot $K\subset {S}^{3}$, let ${u}^{-}\left(K\right)$ (respectively, ${u}^{+}\left(K\right)$) 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 ${\mathfrak{𝔩}}^{-}\left(K\right)$, ${\mathfrak{𝔩}}^{+}\left(K\right)$ and $\mathfrak{𝔩}\left(K\right)$, which give lower bounds on ${u}^{-}\left(K\right)$, ${u}^{+}\left(K\right)$ and the unknotting number $u\left(K\right)$, respectively. The invariant $\mathfrak{𝔩}\left(K\right)$ only vanishes for the unknot, and satisfies $\mathfrak{𝔩}\left(K\right)\ge {\nu }^{+}\left(K\right)$, while the difference $\mathfrak{𝔩}\left(K\right)-{\nu }^{+}\left(K\right)$ can be arbitrarily large. We also present several applications towards bounding the unknotting number, the alteration number and the Gordian distance.

##### Keywords
knot, unknotting number, knot Floer homology, torsion
Primary: 57M27