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On McMullen's and other inequalities for the Thurston norm of link complements

Oliver T Dasbach and Brian S Mangum

Algebraic & Geometric Topology 1 (2001) 321–347

arXiv: math.GT/9911172


In a recent paper, McMullen showed an inequality between the Thurston norm and the Alexander norm of a 3–manifold. This generalizes the well-known fact that twice the genus of a knot is bounded from below by the degree of the Alexander polynomial.

We extend the Bennequin inequality for links to an inequality for all points of the Thurston norm, if the manifold is a link complement. We compare these two inequalities on two classes of closed braids.

In an additional section we discuss a conjectured inequality due to Morton for certain points of the Thurston norm. We prove Morton’s conjecture for closed 3–braids.

Thurston norm, Alexander norm, multivariable Alexander polynomial, fibred links, positive braids, Bennequin's inequality, Bennequin surface, Morton's conjecture
Mathematical Subject Classification 2000
Primary: 57M25
Secondary: 57M27, 57M50
Forward citations
Received: 14 December 2000
Revised: 21 May 2001
Accepted: 25 May 2001
Published: 31 May 2001
Oliver T Dasbach
University of California
Department of Mathematics
Riverside CA 92521-0135
Brian S Mangum
Barnard College
Columbia University
Department of Mathematics
New York NY 10027