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

Keiko Kawamuro

Algebraic & Geometric Topology 6 (2006) 2313–2350

DOI: 10.2140/agt.2006.6.2313

arXiv: 0907.1019


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 L, then it is also true for the (p,q)-cable of K and for the connect sum of K and L.


braids, braid index, Morton-Franks-Williams inequality

Mathematical Subject Classification

Primary: 57M25

Secondary: 57M27

Forward citations

Received: 28 April 2006
Accepted: 21 July 2006
Published: 8 December 2006

Keiko Kawamuro
Math Department
Rice University
6100 S Main St
Houston TX 77005-1892