Crossing information and warping polynomials about the trefoil knot (Q2920400)

In knot theory a knot is usually represented by a planar diagram. Given a knot diagram one can define a Gauss diagram (or a Gauss code) of it. In particular the knot diagram is completely determined by the associated Gauss code. In general a Gauss code is a sequence of O's and U's, a pair \{O, U\} corresponding to a crossing on the diagram, and the sign information of each crossing point also indicated. In [Eur. J. Comb. 20, No. 7, 663--690 (1999; Zbl 0938.57006)], \textit{L. H. Kauffman} discussed when a Gauss code can be realized as a classical knot diagram.NEWLINENEWLINEIn the paper under review, the authors consinder sequences of O's and U's without information of pairing and the signs of the crossings. They name it the OU sequence of a knot diagram. The first question is about the realization of a given OU sequence, here the authors show that the obvious condition that the number of overcrossings is equivalent to the number of undercrossings is sufficient. The main result of this paper is the characterization of OU sequences which are always realized by a trivial knot. On the other hand the OU sequences of the trefoil knot are studied and it is proved that if an OU sequence can be realized by a nontrivial knot then it can always be realized by the trefoil knot. The key idea of the proof is the bridge number, which can be read off from the OU sequence directly. As an application the authors characterize the warping polynomials for all diagrams of the trefoil knot.