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      Seifert vs slice genera of knots in twist families and a characterization of braid axes

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          Abstract

          Twisting a knot \(K\) in \(S^3\) along a disjoint unknot \(c\) produces a twist family of knots \(\{K_n\}\) indexed by the integers. Comparing the behaviors of the Seifert genus \(g(K_n)\) and the slice genus \(g_4(K_n)\) under twistings, we prove that if \(g(K_n) - g_4(K_n) < C\) for some constant \(C\) for infinitely many integers \(n > 0\) or \(g(K_n) / g_4(K_n) \to 1\) as \(n \to \infty\), then either the winding number of \(K\) about \(c\) is zero or the winding number equals the wrapping number. As a key application, if \(\{K_n\}\) or the mirror twist family \(\{\overline{K_n}\}\) contains infinitely many tight fibered knots, then the latter must occur. We further develop this to show that \(c\) is a braid axis of \(K\) if and only if both \(\{K_n\}\) and \(\{\overline{K_n}\}\) each contain infinitely many tight fibered knots. We also give a necessary and sufficient condition for \(\{ K_n \}\) to contain infinitely many L-space knots, and show (modulo a conjecture) that satellite L-space knots are braided satellites.

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          Most cited references27

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          On knot Floer homology and lens space surgeries

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            Khovanov homology and the slice genus

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              Knot Floer homology detects fibred knots

              Yi Ni (2006)
              Ozsv\'ath and Szab\'o conjectured that knot Floer homology detects fibred knots in \(S^3\). We will prove this conjecture for null-homologous knots in arbitrary closed 3--manifolds. Namely, if \(K\) is a knot in a closed 3--manifold \(Y\), \(Y-K\) is irreducible, and \(\hat{HFK}(Y,K)\) is monic, then \(K\) is fibred. The proof relies on previous works due to Gabai, Ozsv\'ath--Szab\'o, Ghiggini and the author. A corollary is that if a knot in \(S^3\) admits a lens space surgery, then the knot is fibred.
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                Author and article information

                Journal
                2017-05-29
                Article
                1705.10373
                e5c5d627-0b44-408c-8e5a-37fa6c0e65cc

                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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                Custom metadata
                Primary 57M25, 57M27, Secondary 57R17, 57R58
                34 pages, 12 figures
                math.GT

                Geometry & Topology
                Geometry & Topology

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