Brendan Owens

Concordance groups of links

We define a notion of concordance based on Euler characteristic, and show that it gives rise to a concordance group of links in the 3-sphere, which has the concordance group of knots as a direct summand with infinitely generated complement. We consider variants of this using oriented and nonoriented surfaces as well as smooth and locally flat embeddings.

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 t0 on Σ(L), the two–fold cover of S3 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 (Σ(L), t0).

On slicing invariants of knots

The slicing number of a knot, u_s(K), is the minimum number of crossing changes required to convert K to a slice knot. This invariant is bounded above by the unknotting number and below by the slice genus g_s(K). We show that for many knots, previous bounds on the unknotting number obtained by Ozsvath and Szabo and by the author in fact give bounds on the slicing number. Livingston defined another invariant U_s(K), which takes into account signs of crossings changed to get a slice knot and which is bounded above by the slicing number and below by the slice genus. We exhibit an infinite family of knots K_n with slice genus n and Livingston invariant greater than n. Our bounds are based on restrictions (using Donaldsons diagonalisation theorem or Heegaard Floer homology) on the intersection forms of four-manifolds bounded by the double branched cover of a knot.

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.

Instantons on cylindrical manifolds and stable bundles

Let be a smooth complex curve, and let S be the product ruled surface CP 1 . We prove a correspondence conjectured by Donaldson between nite energy U (2){instantons over S 1 R, and rank 2 holomorphic bundles over