#### Algebraic and Geometric Topology 1 (2001),
paper no. 16, pages 321-347.

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

### Oliver T. Dasbach, Brian S. Mangum

**Abstract**.
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.
**Keywords**.
Thurston norm, Alexander norm, multivariable Alexander polynomial,
fibred links, positive braids, Bennequin's inequality, Bennequin
surface, Morton's conjecture

**AMS subject classification**.
Primary: 57M25.
Secondary: 57M27, 57M50.

**DOI:** 10.2140/agt.2001.1.321

**E-print:** `arXiv:math.GT/9911172`

Submitted: 14 December 2000.
(Revised: 21 May 2001.)
Accepted: 25 May 2001.
Published: 31 May 2001.

Notes on file formats
Oliver T. Dasbach, Brian S. Mangum

University of California, Riverside, Department of Mathematics

Riverside, CA 92521 - 0135, USA

Barnard College/Columbia University, Department of Mathematics

New York, NY 10027, USA

Email: kasten@math.ucr.edu, mangum@math.columbia.edu

URL: http://www.math.ucr.edu/~kasten

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