On knot Floer homology and lens space surgeries

By Peter S. Ozsv'ath, Zolt'an Szab'o. 2003

482 citations recorded by the source corpus.

Abstract In an earlier paper, we used the absolute grading on Heegaard Floer homology HF + to give restrictions on knots in S 3 which admit lens space surgeries. The aim of the present article is to exhibit stronger restrictions on such knots, arising from knot Floer homology. One consequence is that the non-zero coefficients of the Alexander polynomial of such a knot are ± 1 . This information can in turn be used to prove that certain lens spaces are not obtained as integral surgeries on knots. In fact, combining our results with constructions of Berge, we classify lens spaces L ( p , q ) which arise as integral surgeries on knots in S 3 with | p | ⩽ 1500 . Other applications include bounds on the four-ball genera of knots admitting lens space surgeries (which are sharp for Berge's knots), and a constraint on three-manifolds obtained as integer surgeries on alternating knots, which is closely to related to a theorem of Delman and Roberts.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/04246a5545f87f25ed006696e39a67d8aad59128

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Discrete Mathematics and Combinatorics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

On the Heegaard Floer homology of branched double-covers (2003, background): “jection, then the rank of HFd(Σ(L)) agrees with the number of elements in H2(Σ(L);Z), i.e. that Σ(L) is what might be called an “ungraded Heegaard Floer homology lens space” or, in the terminology of [16], an L-space. (Note that this argument applies to a larger class of links which includes all alt”
Cabling, contact structures and mapping class monoids (2010, background): “In particular, any knot in S3 with a Dehn surgery yielding a lens space is known to be fibered [48], [45].”
Lens space surgeries and L-space homology spheres (2007, methodology): “f section 5 of [29]) We refer to the standard Heegaard diagram of K(p, q, k) described in section 2.”
Knot polynomials and knot homologies (2005, methodology): “The second method applies if K has a lens space, or, more generally an L-space surgery.”
Knot Floer homology, genus bounds, and mutation (2003, background): “to this are the (3,4)-torus knot (which appears in the tables under the name 8 19) and a certain nine-crossing knot 9 42. In fact, the knot Floer homologies of these two knots have been determined in [16] and [18] respectively, where it is shown that: HFK\(8 42,i) ∼= Z (0) if i = 3 Z (−1) if ”
On Floer homology and the Berge conjecture on knots admitting lens space surgeries (2007, background): “6 of [24]) Let (S(3), K) be a knot of Seifert genus g(K) and suppose that there exists an integer p > 0 such that S(3) p(K) is an L-space.”
Knot Floer homology and the four-ball genus (2003, background): “Letting F(K,m) ⊂ ĈF (S3) be the subcomplex generated by intersection points whose filtration level is less than or equal to m, we obtain an induced sequence of maps ιmK : H∗(F(K,m)) −→ H∗(ĈF (S3)) = ĤF (S3) ∼= Z, which are isomorphisms for all sufficiently large integers m.”
Involutive Heegaard Floer homology (2015, background): “In [41], Ozsváth and Szabó proved that if K is an L-space knot, then the Alexander polynomial of K is of the form”
Heegaard Floer homology and genus one, one‐boundary component open books (2008, background): “The following property of Heegaard Floer homology, proved in [36], is very useful for the case d = 1.”
Link Floer homology and the Thurston norm (2006, background): “ave particularly simple Heegaard Floer homology groups – in each Spinc structure, HF+(S3 N(K)) ∼= T +, i.e. these spaces are “Heegaard Floer homology lens spaces”, or L-spaces in the terminology from [18]. As such, the ﬁltered homotopy type of the knot ﬁltration of torus knots is immediately determi”