  We give examples of knots with some unusual properties of the crossing number
of positive diagrams or strand number of positive braid representations. In
particular we show that positive braid knots may not have positive minimal
(strand number) braid representations, giving a counterpart to results of
Franks-Williams and Murasugi. Other examples answer questions of Cromwell on
homogeneous and (partially) of Adams on almost alternating knots.
  We give a counterexample to, and a corrected version of a theorem of Jones on
the Alexander polynomial of 4-braid knots. We also give an example of a knot on
which all previously applied braid index criteria fail to estimate sharply
(from below) the braid index. A relation between (generalizations of) such
examples and a conjecture of Jones that a minimal braid representation has
unique writhe is discussed.
  Finally, we give a counterexample to Morton's conjecture relating the genus
and degree of the skein polynomial.
