Algebraic & Geometric Topology

The algebraic crossing number and the braid index of knots and links

Keiko Kawamuro

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It has been conjectured that the algebraic crossing number of a link is uniquely determined in minimal braid representation. This conjecture is true for many classes of knots and links.

The Morton–Franks–Williams inequality gives a lower bound for braid index. And sharpness of the inequality on a knot type implies the truth of the conjecture for the knot type.

We prove that there are infinitely many examples of knots and links for which the inequality is not sharp but the conjecture is still true. We also show that if the conjecture is true for K and , then it is also true for the (p,q)–cable of K and for the connect sum of K and .

Article information

Algebr. Geom. Topol., Volume 6, Number 5 (2006), 2313-2350.

Received: 28 April 2006
Accepted: 21 July 2006
First available in Project Euclid: 20 December 2017

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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

braids braid index Morton-Franks-Williams inequality


Kawamuro, Keiko. The algebraic crossing number and the braid index of knots and links. Algebr. Geom. Topol. 6 (2006), no. 5, 2313--2350. doi:10.2140/agt.2006.6.2313.

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