Four-genera of links and Heegaard Floer homology

Abstract

For links with vanishing pairwise linking numbers, the link components bound pairwise disjoint surfaces in B 4 . In this paper, we describe the set of genera of such surfaces in terms of the h-function, which is a link invariant from Heegaard Floer homology. In particular, we use the h-function to give lower bounds for the 4-genus of the link. For L-space links, the h-function is explicitly determined by Alexander polynomials of the link and sublinks. We show some L-space links where the lower bounds are sharp, and also describe all possible genera of disjoint surfaces bounded by such links.

$L$-space surgeries on links

“…In Figure 11 Case (II), we suppose that b 1 < 2g 1 − 1 and b 2 > 2g 2 − 1. The argument for regions 1,2,4,12,16,13,14,9,10,11,15 is very similar to the argument in Case (I). For points in regions 5 or 6, we cannot use the surgery induction argument.…”

“…The rest of the argument is the same as the one in the case that l is even. We refer the readers to [10] for more details about maximal lattice points. If there exists a maximal lattice point (s 1 , s 2 ) for L, then b i ≥ s i for i = 1, 2.…”

“…2 ). In [10,Example 4.4], we compute its H-function, and get b 1 = 0 and b 2 = 1. Its Alexander polynomial satisfies the assumptions in Corollary 4.26.…”

“…We claim that the coefficients of t y 1 1 t y 2 2 in∆ L (t 1 , t 2 ) are 0 for all y ≻ s if and only if for all y ′ ≻ (T (s 1 ), s 2 ), the coefficients of t y ′ 1 1 t y ′ 2 2 are 0 in∆ Lp,q (t 1 , t 2 ). The proof can be found in [10,Theorem 4.7], and we don't repeat the argument here. By the claim, the coefficients of the terms t is −1.…”

L‐space surgeries on 2‐component L‐space links

“…(b) If d 1 < 0, d 2 < 0, det(Λ) > 0, then the free part of HF − (S 3 d1,d2 (L), u) is generated by a chain inC 11 (Λ, u).…”