Knot Floer homology and Seifert surfaces

Let $K$ be a knot in $S^{3}$ of genus $g$ and let $n > 0 .$ We show that if $rk \hat{H F K} (K, g) < 2^{n + 1}$ (where $\hat{H F K}$ 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.

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Keywords

Alexander polynomial, Seifert surface, Floer homology

Mathematical Subject Classification 2000

Primary: 57M27, 57R58

References

Publication

Received: 7 December 2007

Revised: 25 February 2008

Accepted: 25 February 2008

Published: 12 May 2008

Authors

Andras Juhasz
Department of Mathematics Princeton University Princeton, NJ 08544 USA

Volume 8, issue 1 (2008)

Andras Juhasz

Abstract