jats:titleAbstract</jats:title>jats:pThe unwinding number of a complex number was introduced to process automatic computations involving complex numbers and multi-valued complex functions, and has been successfully applied to computations involving branches of the Lambert jats:italicW</jats:italic> function. In this partly expository note we discuss the unwinding number from a purely topological perspective, and link it to the classical winding number of a curve in the complex plane. We also use the unwinding number to give a representation of the branches jats:inline-formulajats:alternativesjats:tex-math$$W_k$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:msub mml:miW</mml:mi> mml:mik</mml:mi> </mml:msub> </mml:math></jats:alternatives></jats:inline-formula> of the Lambert jats:italicW</jats:italic> function as a line integral.</jats:p>