The kernel of $Burau(4) \otimes Z_p$ is all pseudo-Anosov

The Burau estimate for the entropy of a braid

The topological entropy of a braid is the infimum of the entropies of all homeomorphisms of the disc which have a finite invariant set represented by the braid. When the isotopy class represented by the braid is pseudo-Anosov or is reducible with a pseudo-Anosov component, this entropy is positive. Fried and Kolev proved that the entropy is bounded below by the logarithm of the spectral radius of the braid's Burau matrix, $B(t)$, after substituting a complex number of modulus~1 in place of $t$. In this paper we show that for a pseudo-Anosov braid the estimate is sharp for the substitution of a…