Results 1  10
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70
Khovanov homology and the slice genus
, 2003
"... Abstract. We use Lee’s work on the Khovanov homology to define a knot invariant s. We show that s(K) is a concordance invariant and that it provides a lower bound for the slice genus of K. As a corollary, we give a purely combinatorial proof of the Milnor conjecture. 1. ..."
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Cited by 86 (5 self)
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Abstract. We use Lee’s work on the Khovanov homology to define a knot invariant s. We show that s(K) is a concordance invariant and that it provides a lower bound for the slice genus of K. As a corollary, we give a purely combinatorial proof of the Milnor conjecture. 1.
Knot Floer Homology and the fourball genus
 Geom. Topol
"... Abstract. We use the knot filtration on the Heegaard Floer complex ĈF to define an integer invariant τ(K) for knots. Like the classical signature, this invariant gives a homomorphism from the knot concordance group to Z. As such, it gives lower bounds for the slice genus (and hence also the unknotti ..."
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Cited by 61 (8 self)
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Abstract. We use the knot filtration on the Heegaard Floer complex ĈF to define an integer invariant τ(K) for knots. Like the classical signature, this invariant gives a homomorphism from the knot concordance group to Z. As such, it gives lower bounds for the slice genus (and hence also the unknotting number) of a knot; but unlike the signature, τ gives sharp bounds on the fourball genera of torus knots. As another illustration, we use calculate the invariant for several tencrossing knots. 1.
A combinatorial description of knot Floer homology
, 2006
"... Given a grid presentation of a knot (or link) K in the threesphere, we describe a Heegaard diagram for the knot complement in which the Heegaard surface is a torus and all elementary domains are squares. Using this diagram, we obtain a purely combinatorial description of the knot Floer homology of ..."
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Cited by 57 (18 self)
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Given a grid presentation of a knot (or link) K in the threesphere, we describe a Heegaard diagram for the knot complement in which the Heegaard surface is a torus and all elementary domains are squares. Using this diagram, we obtain a purely combinatorial description of the knot Floer homology of K.
On knot Floer homology and lens space surgery
"... 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 th ..."
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Cited by 55 (12 self)
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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 all the nonzero coefficients of the Alexander polynomial of such a knot are ±1. This information in turn can 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 fourball genera of knots admitting lens space surgeries (which are sharp for Berge’s knots), and a constraint on threemanifolds obtained as integer surgeries on alternating knots, which is closely to related to a theorem of Delman and Roberts. 1.
Qusipositivity as an obstruction to sliceness
 Bull. Amer. Math. Soc. (N.S
, 1993
"... Abstract. For an oriented link L ⊂ S 3 = ∂D 4, let χs(L) be the greatest Euler characteristic χ(F) of an oriented 2manifold F (without closed components) smoothly embedded in D 4 with boundary L. A knot K is slice if χs(K) = 1. Realize D 4 in C 2 as {(z, w) : z  2 + w  2 ≤ 1}. It has been conj ..."
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Cited by 37 (1 self)
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Abstract. For an oriented link L ⊂ S 3 = ∂D 4, let χs(L) be the greatest Euler characteristic χ(F) of an oriented 2manifold F (without closed components) smoothly embedded in D 4 with boundary L. A knot K is slice if χs(K) = 1. Realize D 4 in C 2 as {(z, w) : z  2 + w  2 ≤ 1}. It has been conjectured that, if V is a nonsingular complex plane curve transverse to S 3, then χs(V ∩ S 3) = χ(V ∩ D 4). Kronheimer and Mrowka have proved this conjecture in the case that V ∩ D 4 is the Milnor fiber of a singularity. I explain how this seemingly special case implies both the general case and the “sliceBennequin inequality ” for braids. As applications, I show that various knots are not slice (e.g., pretzel knots like P(−3, 5, 7); all knots obtained from a positive trefoil O{2,3} by iterated untwisted positive doubling). As a sidelight, I give an optimal counterexample to the “topologically locallyflat Thom conjecture”. 1. A brief history of sliceness A link is a compact 1manifold without boundary L (i.e., finite union of simple closed curves) smoothly embedded in the 3sphere S 3; a knot is a link with one component. If S 3 is realized in R 4 as, say, the unit sphere, then a natural way to construct links is to intersect suitable twodimensional subsets X ⊂ R 4 with S 3; one may then ask how constraints on X are reflected in constraints on the link X ∩ S 3. For instance, Fox and Milnor (c. 1960) considered, in effect, the case that X is a smooth 2sphere intersecting S 3 transversally; at Moise’s suggestion, Fox [5] adopted the adjective slice to describe the knots and links X ∩ S 3 so constructed. Fox and Milnor [6] gave a criterion for a knot K to be slice: its Alexander polynomial ∆K(t) ∈ Z[t, t −1] must have the form F(t)F(t −1). This shows that, for instance, the two trefoil knots O{2, ±3} are not slice (since ∆ O{2,±3} = t −1 −1+t is not of the form F(t)F(t −1)), but it says nothing about the two granny knots O{2, 3} O{2, 3}, O{2, −3} O{2, −3} (indeed, both granny knots share the Alexander polynomial
Computations of the Ozsváth–Szabó knot concordance invariant
, 2004
"... Ozsváth and Szabó have defined a knot concordance invariant τ that bounds the 4–ball genus of a knot. Here we discuss shortcuts to its computation. We include examples of Alexander polynomial one knots for which the invariant is nontrivial, including all iterated untwisted positive doubles of knots ..."
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Cited by 34 (5 self)
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Ozsváth and Szabó have defined a knot concordance invariant τ that bounds the 4–ball genus of a knot. Here we discuss shortcuts to its computation. We include examples of Alexander polynomial one knots for which the invariant is nontrivial, including all iterated untwisted positive doubles of knots with nonnegative Thurston–Bennequin number, such as the trefoil, and explicit computations for several 10 crossing knots. We also note that a new proof of the Slice–Bennequin Inequality quickly follows from these techniques.
Monopoles and lens space surgeries
 ArXive:math.GT/0310164
, 2003
"... Abstract. Monopole Floer homology is used to prove that real projective threespace cannot be obtained from Dehn surgery on a nontrivial knot in the threesphere. To obtain this result, we use a surgery long exact sequence for monopole Floer homology, together with a nonvanishing theorem, which sh ..."
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Cited by 34 (10 self)
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Abstract. Monopole Floer homology is used to prove that real projective threespace cannot be obtained from Dehn surgery on a nontrivial knot in the threesphere. To obtain this result, we use a surgery long exact sequence for monopole Floer homology, together with a nonvanishing theorem, which shows that monopole Floer homology detects the unknot. In addition, we apply these techniques to give information about knots which admit lens space surgeries, and to exhibit families of threemanifolds which do not admit taut foliations. 1.
Legendrian knots, transverse knots and combinatorial Floer homology
, 2008
"... Using the combinatorial approach to knot Floer homology, we define an invariant for Legendrian knots (or links) in the threesphere, with values in knot Floer homology. This invariant can also be used to construct an invariant of transverse knots. ..."
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Cited by 21 (6 self)
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Using the combinatorial approach to knot Floer homology, we define an invariant for Legendrian knots (or links) in the threesphere, with values in knot Floer homology. This invariant can also be used to construct an invariant of transverse knots.
Knots with unknotting number one and Heegaard Floer homology
"... Abstract. We use Heegaard Floer homology to give obstructions to unknotting a knot with a single crossing change. These restrictions are particularly useful in the case where the knot in question is alternating. As an example, we use them to classify all knots with crossing number less than or equal ..."
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Cited by 18 (2 self)
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Abstract. We use Heegaard Floer homology to give obstructions to unknotting a knot with a single crossing change. These restrictions are particularly useful in the case where the knot in question is alternating. As an example, we use them to classify all knots with crossing number less than or equal to nine and unknotting number equal to one. We also classify alternating knots with ten crossings and unknotting number equal to one. 1.