76 research outputs found
Equivariant embeddings of rational homology balls
We generalise theorems of Khodorovskiy and Park-Park-Shin, and give new
topological proofs of those theorems, using embedded surfaces in the 4-ball and
branched double covers. These theorems exhibit smooth codimension-zero
embeddings of certain rational homology balls bounded by lens spaces.Comment: 27 pages, 25 figures. V2: Improved exposition incorporating referee's
suggestions. Accepted for publication in Q. J. Math. V3: minor correction
Unlinking information from 4-manifolds
We generalise theorems of Cochran-Lickorish and Owens-Strle to the case of
links with more than one component. This enables the use of linking forms on
double branched covers, Heegaard Floer correction terms, and Donaldson's
diagonalisation theorem to complete the table of unlinking numbers for nonsplit
prime links with crossing number nine or less.Comment: 18 pages, 2 figures. V2: Improved exposition incorporating referee's
suggestions. Accepted for publication in Bull. London Math. So
Signatures, Heegaard Floer correction terms and quasi-alternating links
Turaev showed that there is a well-defined map assigning to an oriented link
L in the three-sphere a Spin structure t_0 on Sigma(L), the 2-fold cover of S^3
branched along L. We prove, generalizing results of Manolescu-Owens and
Donald-Owens, that for an oriented quasi-alternating link L the signature of L
equals minus four times the Heegaard Floer correction term of (Sigma(L), t_0).Comment: V2: Improved exposition incorporating referee's suggestions; 3
figures, 6 pages. Accepted for publication by the Proceedings of the American
Mathematical Societ
Immersed disks, slicing numbers and concordance unknotting numbers
We study three knot invariants related to smoothly immersed disks in the
four-ball. These are the four-ball crossing number, which is the minimal number
of normal double points of such a disk bounded by a given knot; the slicing
number, which is the minimal number of crossing changes to a slice knot; and
the concordance unknotting number, which is the minimal unknotting number in a
smooth concordance class. Using Heegaard Floer homology we obtain bounds that
can be used to determine two of these invariants for all prime knots with
crossing number ten or less, and to determine the concordance unknotting number
for all but thirteen of these knots. As a further application we obtain some
new bounds on Gordian distance between torus knots. We also give a strengthened
version of Ozsvath and Szabo's obstruction to unknotting number one.Comment: 24 pages, 5 figures. V2: added section on Gordian distances between
torus knots. V3: Improved exposition incorporating referees' suggestions.
Accepted for publication in Comm. Anal. Geo
Dehn surgeries and negative-definite four-manifolds
Given a knot <i>K</i> in the three-sphere, we address the question: Which Dehn surgeries on <i>K</i> bound negative-definite four-manifolds? We show that the answer depends on a number <i>m(K)</i>, which is a smooth concordance invariant. We study the properties of this invariant and compute it for torus knots
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