Polynomials, Binary Trees, and Positive Braids

Abstract

In knot theory, a common task is to take a given knot diagram and generate from it a polynomial. One method for accomplishing this is to employ a skein relation to convert the knot into a type of labeled binary tree and from this tree derive a two-variable polynomial. The purpose of this paper is to determine, in a simplified setting, which polynomials can be generated from labeled binary trees. We give necessary and sufficient conditions for a polynomial to be constructible in this fashion and we will provide a method for reconstructing the generating tree from such a polynomial. We conclude with an application of this theorem to a class of knots and links given by closed positive braids.