On the bridge number of knot diagrams with minimal crossings

Details On the bridge number of knot diagrams with minimal crossings On the bridge number of knot diagrams with minimal crossings

2003-01-28

arxiv.org/abs/math/0301320v1

Mathematics

Geometric Topology

18 pages, 7 figures

Scientific paper

10.1017/S0305004104007753

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 shape 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.

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