On the crossing number of positive knots and braids and braid index criteria of Jones and Morton-Williams-Franks

摘要

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