PN sequences obtained from signs of crossings of knot diagrams

Abstract

For an oriented knot diagram, we define a cyclic word in letters [Formula: see text] and [Formula: see text] corresponding to positive and negative crossings along the diagram, respectively. We give a necessary and sufficient condition for a cyclic word in [Formula: see text] and [Formula: see text] to be obtained from some knot diagram. Also, we prove that any PN sequence of a diagram of a nontrivial knot contains at least four subsequences [Formula: see text] and [Formula: see text].