On the genus of thin knots. I came across this papper by JA Baldwin which presents a combinatorial definition for the knot Floer homology. At a certain paragraph of the third page the author makes the next statement: the genus of a homologically thin knot is the degree of its Alexander polynomial. I just want to make sure that the author does mean the genus I know, namely the minimal genus taken over all spaning surfaces of the knot. Thanks in advance.

In this context, genus is the minimal genus taken over all Seifert surfaces of the knot (i.e. over all oriented spanning surfaces of the knot). Ozsvath and Szabo prove (in this paper) that the genus of a knot is the maximum Alexander grading in which the knot Floer homology is non-trivial. If the knot Floer homology of a knot is thin, then the degree of the Alexander polynomial is equal to the maximum Alexander grading in which the knot Floer homology is non-trivial (and thus it is also equal to the genus of the knot).