Algebraic & Geometric Topology

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

Oliver T Dasbach and Brian S Mangum

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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.

Article information

Algebr. Geom. Topol., Volume 1, Number 1 (2001), 321-347.

Received: 14 December 2000
Revised: 21 May 2001
Accepted: 25 May 2001
First available in Project Euclid: 21 December 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}
Secondary: 57M27: Invariants of knots and 3-manifolds 57M50: Geometric structures on low-dimensional manifolds

Thurston norm Alexander norm multivariable Alexander polynomial fibred links positive braids Bennequin's inequality Bennequin surface Morton's conjecture


Dasbach, Oliver T; Mangum, Brian S. On McMullen's and other inequalities for the Thurston norm of link complements. Algebr. Geom. Topol. 1 (2001), no. 1, 321--347. doi:10.2140/agt.2001.1.321.

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