  We continue the study of the genus of knot diagrams, deriving a new
description of generators using Hirasawa's algorithm. This description leads to
good estimates on the maximal number of crossings of generators and allows us
to complete their classification for knots of genus 4.
  As applications of the genus 4 classification, we establish non-triviality of
the skein polynomial on $k$-almost positive knots for $k\le 4$, and of the
Jones polynomial for $k\le 3$. For $k\le 4$, we classify the occurring achiral
knots, and prove a trivializability result for $k$-almost positive unknot
diagrams. This yields also estimates on the number of unknotting Reidemeister
moves. We describe the positive knots of signature (up to) 4.
  Using a study of the skein polynomial, we prove the exactness of the
Morton-Williams-Franks braid index inequality and the existence of a minimal
string Bennequin surface for alternating knots up to genus 4. We also prove for
such knots conjectures of Hoste and Fox about the roots and coefficients of the
Alexander polynomial.
