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On the bridge number of knot diagrams with minimal crossings

Published online by Cambridge University Press:  02 November 2004

JAE-WOOK CHUNG  and
XIAO-SONG LIN
JAE-WOOK CHUNG
Affiliation:
Department of Mathematics, University of California, Riverside, CA 92521, U.S.A. e-mail: [email protected]@math.ucr.edu
XIAO-SONG LIN
Affiliation:
Department of Mathematics, University of California, Riverside, CA 92521, U.S.A. e-mail: [email protected]@math.ucr.edu
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Abstract

Given a diagram $D$ of a knot $K$, we consider the number $c(D)$ of crossings and the number $b(D)$ of overpasses of $D$. We show that, if $D$ is a diagram of a nontrivial knot $K$ whose number $c(D)$ of crossings is minimal, then $1+\sqrt{1+c(D)} \leq b(D)\leq c(D)$. These inequalities are sharp in the sense that the upper bound of $b(D)$ is achieved by alternating knots and the lower bound of $b(D)$ is achieved by torus knots. The second inequality becomes an equality only when the knot is an alternating knot. We prove that the first inequality becomes an equality only when the knot is a torus knot.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 137 , Issue 3 , November 2004 , pp. 617 - 632
Copyright
© 2004 Cambridge Philosophical Society

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