
Let us define $$g(L)$$ as the minimal genus of a Seifert surface of $$L$$ (where the genus of a disconnected surface is defined as the sum of the genera of the components). For example, $$g(U\sqcup U) = g(T_{2,2}) = 0$$. While there are some inequalities between $$\chi(L)$$ and $$g(L)$$, they do not generally determine each other.

Can one also read off $$g(L)$$ from $$\wHFK(L)$$ for all links $$L$$? How about link Floer homology $$\wHFL(L)$$?

More generally, we may consider a link $$L$$ with a fixed partition of its components into sublinks $$L_1, \ldots, L_k$$, and consider Seifert surfaces $$\Sigma$$ that have $$k$$ connected components $$\Sigma_1, \ldots, \Sigma_k$$, with $$\Sigma_i$$ a connected Seifert surface for $$L_i$$ for all $$i\in\{1,\ldots, k\}$$. Does $$\wHFL(L)$$ detect whether such a $$\Sigma$$ exists, and if so, what the maximal Euler characteristic of such a $$\Sigma$$ is?

A special case of this last question is: does $$\wHFL(L)$$ of a link $$L$$ detect whether $$L$$ is a boundary link? ($$L$$ is called a boundary link if $$L$$ is the boundary of a Seifert surface all of whose connected components have only one boundary component.)

• Do you have a reference for "Ni has shown that..."? I believe you, I just want to look at the paper myself! Mar 17, 2022 at 13:53
• Presumably, A note on knot Floer homology of links. Geom. Topol. 10 (2006), 695–713. Mar 17, 2022 at 16:12
• @DannyRuberman Thanks! Mar 18, 2022 at 18:21