Volume 8, issue 1 (2008)

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Knot Floer homology and Seifert surfaces

Andras Juhasz

Algebraic & Geometric Topology 8 (2008) 603–608
Abstract

Let $K$ be a knot in ${S}^{3}$ of genus $g$ and let $n>0.$ We show that if $rk\stackrel{̂}{HFK}\left(K,g\right)<{2}^{n+1}$ (where $\stackrel{̂}{HFK}$ denotes knot Floer homology), in particular if $K$ is an alternating knot such that the leading coefficient ${a}_{g}$ of its Alexander polynomial satisfies $|{a}_{g}|<{2}^{n+1},$ then $K$ has at most $n$ pairwise disjoint nonisotopic genus $g$ Seifert surfaces. For $n=1$ this implies that $K$ has a unique minimal genus Seifert surface up to isotopy.

Keywords
Alexander polynomial, Seifert surface, Floer homology
Mathematical Subject Classification 2000
Primary: 57M27, 57R58