## Algebraic & Geometric Topology

### Bridge number and Conway products

Ryan C Blair

#### Abstract

In this paper, we give a structure theorem for c-incompressible Conway spheres in link complements in terms of the standard height function on $S3$. We go on to define the generalized Conway product $K1∗cK2$ of two links $K1$ and $K2$. Provided $K1∗cK2$ satisfies minor additional hypotheses, we prove the lower bound $β(K1∗cK2)≥β(K1)−1$ for the bridge number of the generalized Conway product where $K1$ is the distinguished factor. Finally, we present examples illustrating that this lower bound is tight.

#### Article information

Source
Algebr. Geom. Topol., Volume 10, Number 2 (2010), 789-823.

Dates
Revised: 15 December 2009
Accepted: 30 January 2010
First available in Project Euclid: 19 December 2017

https://projecteuclid.org/euclid.agt/1513715114

Digital Object Identifier
doi:10.2140/agt.2010.10.789

Mathematical Reviews number (MathSciNet)
MR2629764

Zentralblatt MATH identifier
1200.57003

#### Citation

Blair, Ryan C. Bridge number and Conway products. Algebr. Geom. Topol. 10 (2010), no. 2, 789--823. doi:10.2140/agt.2010.10.789. https://projecteuclid.org/euclid.agt/1513715114

#### References

• J H Conway, An enumeration of knots and links, and some of their algebraic properties, from: “Computational Problems in Abstract Algebra (Proc. Conf., Oxford, 1967)”, Pergamon, Oxford (1970) 329–358
• W B R Lickorish, Prime knots and tangles, Trans. Amer. Math. Soc. 267 (1981) 321–332
• M Scharlemann, Thin position in the theory of classical knots, from: “Handbook of knot theory”, (W Menasco, M Thistlethwaite, editors), Elsevier, Amsterdam (2005) 429–459
• M Scharlemann, M Tomova, Conway products and links with multiple bridge surfaces, Michigan Math. J. 56 (2008) 113–144
• H Schubert, Über eine numerische Knoteninvariante, Math. Z. 61 (1954) 245–288
• J Schultens, Additivity of bridge numbers of knots, Math. Proc. Cambridge Philos. Soc. 135 (2003) 539–544