Adam Simon Levine

A combinatorial spanning tree model for knot Floer homology

Abstract We iterate Manolescu’s unoriented skein exact triangle in knot Floer homology with coefficients in the field of rational functions over Z / 2 Z . The result is a spectral sequence which converges to a stabilized version of δ -graded knot Floer homology. The ( E 2 , d 2 ) page of this spectral sequence is an algorithmically computable chain complex expressed in terms of spanning trees, and we show that there are no higher differentials. This gives the first combinatorial spanning tree model for knot Floer homology.

Knot doubling operators and bordered Heegaard Floer homology

We use bordered Heegaard Floer homology to compute the τ invariant of a family of satellite knots obtained via twisted infection along two components of the Borromean rings, a generalization of Whitehead doubling. We show that τ of the resulting knot depends only on the two twisting parameters and the values of τ for the two companion knots. We also include some notes on bordered Heegaard Floer homology that may serve as a useful introduction to the subject.

Slicing mixed Bing–Whitehead doubles

We show that if K is any knot whose Ozsvath-Szabo concordance invariant tau(K) is positive, the all-positive Whitehead double of any iterated Bing double of K is topologically but not smoothly slice. We also show that the all-positive Whitehead double of any iterated Bing double of the Hopf link (e.g., the all-positive Whitehead double of the Borromean rings) is not smoothly slice; it is not known whether these links are topologically slice.

Splicing knot complements and bordered floer homology

We show that the integer homology sphere obtained by splicing two nontrivial knot complements in integer homology sphere L-spaces has Heegaard Floer homology rank strictly greater than one. In particular, splicing the complements of nontrivial knots in the 3-sphere never produces an L-space. The proof uses bordered Floer homology.

Computing knot Floer homology in cyclic branched covers

We use grid diagrams to give a combinatorial algorithm for computing the knot Floer homology of the pullback of a knot K in its m-fold cyclic branched cover Sigma^m(K), and we give computations when m=2 for over fifty three-bridge knots with up to eleven crossings.

Nonorientable surfaces in homology cobordisms

We investigate constraints on embeddings of a nonorientable surface in a 4–manifold with the homology of M I , where M is a rational homology 3–sphere. The constraints take the form of inequalities involving the genus and normal Euler class of the surface, and either the Ozsvath–Szabo d –invariants or Atiyah–Singer – invariants of M . One consequence is that the minimal genus of a smoothly embedded surface in L.2k; q/ I is the same as the minimal genus of a surface in L.2k; q/ . We also consider embeddings of nonorientable surfaces in closed 4–manifolds.

Khovanov homology and knot Floer homology for pointed links

A well-known conjecture states that for any

Strong Heegaard diagrams and strong L–spaces

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ON KNOTS WITH INFINITE SMOOTH CONCORDANCE ORDER

-component link

Strong L-spaces and left-orderability

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