Knot Floer homology and the unknotting number

Alishahi, Akram¹ ; Eftekhary, Eaman²

¹ Department of Mathematics, University of Georgia, Athens, GA, United States
² School of Mathematics, Institute for Research in Fundamental Sciences (IPM), Tehran, Iran

Geometry & topology, Tome 24 (2020) no. 5, pp. 2435-2469.

Résumé

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.

Détail

DOI : 10.2140/gt.2020.24.2435

Classification : 57M27
Keywords: knot, unknotting number, knot Floer homology, torsion

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