This paper proves a Khovanov-Rozansky analog of Rasmussen’s lower bound for slice genus.
In [Fundam. Math. 199, 1–91 (2008; Zbl 1145.57009)], M. Khovanov and L. Rozansky defined a sequence of graded link homologies which categorify the quantum \(sl_n\)-link polynomials for every positive integer \(n\). The construction uses matrix factorizations theory for potentials of the form \(x^{n+1}\). In [Note on Khovanov link homology, math.QA/0402266, preprint], B. Gornik extended the construction to some non homogeneous potentials. This perturbed Khovanov-Rozansky homology is then filtrated instead of graded and B. Gornik gave an explicit set of generators for it. The case \(n=2\) was already studied by E. S. Lee in [Adv. Math. 197, No. 2, 554–586 (2005; Zbl 1080.57015)] and led to a lower bound for the slice genus given by J. Rasmussen [Khovanov homology and slice genus, Invent. Math., in press].
In the present paper, the author extends Gornik’s construction to even more general potentials. Because of the complexity generated by the Reidemeister III move, he does not prove any topological invariance property of his invariant, but thanks to a particular decomposition of cobordisms in \(\mathbb{R}^4\) into elementary cobordisms, he reaches all the same an analog of Rasmussen’s lower bound for the slice genus.
In section 2, the author recalls Gornik’s construction, focussing on the modifications needed to make it work for any potential \(w\) such that \(\partial_xw(x)\) factorizes through a product of distinct linear factors. He also describes generators for the resulting homology.
Sections 3 and 4 are computational. They contain explicit maps between the perturbed Khovanov-Rozansky homologies of link diagrams which differ by a Morse, a Reidemeister I or a Reidemeister II move. The author also computes the images of the different generators through these maps.
Section 5 is devoted to a refinement of the normal form for closed surfaces in \(\mathbb{R}^4\) given in [A. Kawauchi, T. Shibuya and S. Suzuki, Math. Semin. Notes, Kobe Univ. 10, 75–126 (1982; Zbl 0506.57014)] which avoid Reidemeister III move by replacing them with Reidemeister II moves, saddles and some well controlled punctures.
In the final section 6, the author considers the specific diagram of the unlink occuring in the above refined normal form for surfaces. He computes the grading of a given generator and then follows [J. Rasmussen, loc. cit.] to prove the slice genus lower bound. As an application, he shows that the “\(n\geq3\)” bounds lead to the same proof of the Milnor conjecture on the slice genus of Torus knots as the Rasmussen “\(n=2\)” one.
It should be noted that similar results were proved simultaneously and independently by H. Wu in [Adv. Math. 221, No. 1, 54–139 (2009; Zbl 1167.57007)]. The techniques are however different and involve topological invariance.