## Algebraic & Geometric Topology

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

Keiko Kawamuro

#### Abstract

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 $ℒ.$

#### Article information

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

Dates
Accepted: 21 July 2006
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.agt/1513796639

Digital Object Identifier
doi:10.2140/agt.2006.6.2313

Mathematical Reviews number (MathSciNet)
MR2286028

Zentralblatt MATH identifier
1128.57007

#### Citation

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. https://projecteuclid.org/euclid.agt/1513796639

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