On the Khovanov and knot Floer homologies of quasi-alternating links

By Ciprian Manolescu, Peter S. Ozsv'ath. 2007

180 citations recorded by the source corpus.

Quasi-alternating links are a natural generalization of alternating links. In this paper, we show that quasi-alternating links are "homologically thin" for both Khovanov homology and knot Floer homology. In particular, their bigraded homology groups are determined by the signature of the link, together with the Euler characteristic of the respective homology (i.e. the Jones or the Alexander polynomial). The proofs use the exact triangles relating the homology of a link with the homologies of its two resolutions at a crossing.

Subfields: Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/6b061b50fcbbfaad70171d777eb0f42d12d0e739

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Citations in the corpus

On knot Floer width and Turaev genus (2007, unknown): “G T ] 5 S ep 2”
A survey on Heegaard Floer homology and concordance (2015, result): “r a quasi-alternating knot K, we have that τ(K) = −σ(K) 2 (see Section 3.1 for the deﬁnition of τ) and that HFK\(K) is supported in a single diagonal with respect to the Alexander and Maslov gradings [MO08]. This uniquely speciﬁes HFK\(K), and Petkova [Pet13, Theorem 4] further shows that this uniqu”
Moving basepoints and the induced automorphisms of link Floer homology (2011, background): “All quasialternating knots are –thin [10; 5] and, out of the 85 prime knots up to nine crossings, 83 are quasi-alternating.”
Grid Homology for Knots and Links (2015, unknown): “knot Floer homology for quasi-alternating links [134].”
On the Colored Jones Polynomial, Sutured Floer homology, and Knot Floer homology (2008, background): “) (3) Both theories are “uninteresting” (determined by classical invariants) on quasialternating knots [14].”
The pillowcase and perturbations of traceless representations of knot groups (2013, background): “[30] C. Manolescu and P.Ozsváth, On the Khovanov and knot Floer homologies of quasi-alternating links.”
A combinatorial spanning tree model for knot Floer homology (2011, methodology): “ve crossing; the proof for a negative crossing is extremely similar. First, suppose that the projections DI0, DI1, and DI2 are connected. In this case, we use an argument due to Manolescu and Ozsvath [28]. Choose some Kauﬀman generators x0 ∈ Tα∩ Tη(I0) and x1 ∈ Tα∩ Tη(I1). As in [28, Section 3.5], w”
Four-ball genus bounds and a refinement of the Ozsvath-Szabo tau-invariant (2014, background): “ commutative diagram Ab k −−−−→jA A+ k bvk y v+ k y Bb −−−−→jB B+. The ν+ invariant can be computed explicitly for quasi-alternating knots, a generalization of alternating knots introduced in [MO08]. In fact, Theorem 2 states that ν+ is completely determined by the signature of the knot, just as”
Heegaard Floer homology and genus one, one‐boundary component open books (2008, background): “Specifically, Manolescu and Ozsváth show in [25] that if K is quasi-alternating, then both its reduced Khovanov homology, K̃h(K), and its knot Floer homology, ĤFK(S3 ,K), are σ-thin; that is, both theories are supported”
A link surgery spectral sequence in monopole Floer homology (2009, background): “In fact, K̃h(L) is supported on the single diagonal δ = σ(L)/2 whenever L is quasialternating [17].”
Khovanov homology is an unknot-detector (2010, background): “For alternating (and more generally, quasi-alternating) knots and links, it is known that the rank of the reduced Khovanov cohomology (over Q or over the field of 2 elements) is equal to the lower bound which the above corollary provides [25, 27].”
Involutive Heegaard Floer homology (2015, background): “A more general class of knots, called quasi-alternating knots, were also found to be thin [27].”