Algebraic & Geometric Topology

Bridge number and Conway products

Ryan C Blair

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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 K1cK2 of two links K1 and K2. Provided K1cK2 satisfies minor additional hypotheses, we prove the lower bound β(K1cK2)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
Received: 13 May 2009
Revised: 15 December 2009
Accepted: 30 January 2010
First available in Project Euclid: 19 December 2017

Permanent link to this document
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

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

Keywords
bridge position knot Conway product

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


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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