4
$\begingroup$

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.

$\endgroup$
2
  • $\begingroup$ The genus as you mean it (minimal genus of spanning surface) is at least half of the degree of the Alexander polynomial and in many cases (e.g., fibered knots) it is actually equal to half of the degree. $\endgroup$
    – ThiKu
    Aug 15, 2017 at 11:24
  • $\begingroup$ In the referenced paper (and most papers on knot Floer homology), the Alexander polynomial is symmetrized, so that $\Delta_K(t) = \Delta_K(t^{-1})$. So the degree of the symmetrized Alexander polynomial is half of the degree of the Alexander polynomial when it is written with lowest degree term having degree zero. $\endgroup$ Aug 16, 2017 at 14:03

1 Answer 1

7
$\begingroup$

The genus referred to in the above paper by Baldwin and Levine (and in many related papers in knot Floer homology) is the Seifert genus, also known as the three-genus $g(K)$. This is the minimal genus over all Seifert surfaces bounded by the knot.

The statement that the genus of a thin knot is the degree of its Alexander polynomial is a consequence of two major theorems, both due to Ozsváth and Szabó. The first [1] is that knot Floer homology categorifies the Alexander polynomial and and the second [2] is that it detects the (Seifert) genus of a knot, or more generally the Thurston norm of a three-manifold. In particular, the Seifert genus $g(K)$ is the largest grading $s$ (an integer) where the knot Floer group $\widehat{HFK}_n(K, s) \neq 0$. For homologically thin knots, this integer is also the maximum degree of the symmetrized Alexander polynomial.

Note that the Seifert genus $g(K)$ is not quite, as you said, "the minimal genus taken over all spanning surfaces of the knot" because spanning surfaces may be orientable or non-orientable, whereas Seifert surfaces are necessarily orientable. (The `genus' of a non-orientable surface would be the the minimal number of projective planes required.) All the above statements pertain to orientable surfaces. The smooth four-ball genus and smooth four-dimensional crosscap number also shows in up Heegaard Floer homology too, though I think people are pretty explicit about those being smooth, four-dimensional and/or unoriented.

[1] Ozsváth, Peter; Szabó, Zoltán, Holomorphic disks and knot invariants, Adv. Math. 186, No. 1, 58-116 (2004). ZBL1062.57019.

[2] Ozsváth, Peter; Szabó, Zoltán, Holomorphic disks and genus bounds, Geom. Topol. 8, 311-334 (2004). ZBL1056.57020.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.