On Simplification of the Combinatorial Link Floer Homology

We give a new combinatorial construction of the hat version of the link Floer homology over Z/2Z and verify that in many examples, our complex is smaller than Manolescu–Ozsváth–Sarkar one. Introduction Knot Floer homology is a powerful knot invariant constructed by Ozsváth–Szabo [11] and Rasmussen [14]. In its basic form, the knot Floer homology ĤFK(K) of a knot K ∈ S is a finite–dimensional bigraded vector space over F = Z/2Z ĤFK(K) = ⊕ d∈Z,i∈Z ĤFKd(K, i) , where d is the Maslov and i is the Alexander grading. Its graded Euler characteristic ∑ d,i (−1)rank ĤFKd(K, i)t i = ∆K(t) is equal to the symmetrized Alexander polynomial ∆K(t). The knot Floer homology enjoys the following symmetry extending that of the Alexander polynomial. (1) ĤFKd(K, i) = ĤFKd−2i(K,−i) By the result of Ozsváth–Szabo [10], the maximal Alexander grading i, such that ĤFK∗(K, i) 6= 0 is the Seifert genus g(K) of K. Moreover, Yi Ni showed [7], that the knot is fibered if and only if rank ĤFK∗(K, g(K)) = 1. A concordance invariant bounding from below the slice genus of the knot can also be extracted from knot Floer homology [9]. For torus knots the bound is sharp, providing the new proof of Milnor conjecture. A purely combinatorial proof of the Milnor conjecture was given by Rasmussen in [15] by using Khovanov homology [3]. The knot Floer homology was extended to links in [13]. The first combinatorial construction of the link Floer Homology was given in [5] over F and then in [6] over Z. Both constructions use grid diagrams of links.