## Polynomials, binary trees, and positive braids.(English)Zbl 1297.57025

Summary: 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.

### MSC:

 57M25 Knots and links in the $$3$$-sphere (MSC2010) 05C05 Trees 05C31 Graph polynomials

### Keywords:

binary trees; skein relations; knot polynomials; positive braids
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### References:

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