Volume 8, issue 2 (2008)

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On knot Floer width and Turaev genus

Adam M Lowrance

Algebraic & Geometric Topology 8 (2008) 1141–1162

To each knot K S3 one can associate with its knot Floer homology HFK̂(K), a finitely generated bigraded abelian group. In general, the nonzero ranks of these homology groups lie on a finite number of slope one lines with respect to the bigrading. The width of the homology is, in essence, the largest horizontal distance between two such lines. Also, for each diagram D of K there is an associated Turaev surface, and the Turaev genus is the minimum genus of all Turaev surfaces for K. We show that the width of knot Floer homology is bounded by Turaev genus plus one. Skein relations for genus of the Turaev surface and width of a complex that generates knot Floer homology are given.

knot, Floer, Turaev genus, graphs on surfaces, ribbon graph, width
Mathematical Subject Classification 2000
Primary: 57R58, 57M25
Received: 12 October 2007
Revised: 5 March 2008
Accepted: 25 March 2008
Published: 25 July 2008
Adam M Lowrance
Department of Mathematics
Louisiana State University
Baton Rouge
LA 70803