John A. Baldwin

COMPUTATIONS OF HEEGAARD-FLOER KNOT HOMOLOGY

We compute the knot Floer homology of knots with at most 12 crossings, as well as the τ invariant for knots with at most 11 crossings, using the combinatorial approach described by Manolescu, Ozsvath and Sarkar. We review their construction, giving two examples that can be workout out by hand, and we explain some ideas we used to simplify the computation. We conclude with a discussion of knot Floer homology for small knots, and we formulate a conjecture about the behavior of knot Floer homology under mutation, paying especially close attention to the Kinoshita–Terasaka knot and its Conway mutant. Finally, we discuss a conjecture of Rasmussen on relationship between Khovanov homology and knot Floer homology, and observe that it is consistent with our calculations.

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.

Heegaard Floer homology and genus one, one-boundary component open books

We compute the Heegaard Floer homology of any rational homology 3-sphere with an open book decomposition of the form (T, ϕ), where T is a genus one surface with one-boundary component. In addition, we compute the Heegaard Floer homology of every T 2 -bundle over S 1 with first Betti number equal to 1, and we compare our results with those of Lebow on the embedded contact homology of such torus bundles. We use these computations to place restrictions on Stein-fillings of the contact structures compatible with such open books, to narrow down somewhat the class of 3-braid knots with finite concordance order, and to identify all quasi-alternating links with braid index at most 3.

On the equivalence of Legendrian and transverse invariants in knot Floer homology

Using the grid diagram formulation of knot Floer homology, Ozsvath, Szabo and Thurston defined an invariant of transverse knots in the tight contact 3‐sphere. Shortly afterwards, Lisca, Ozsvath, Stipsicz and Szabo defined an invariant of transverse knots in arbitrary contact 3‐manifolds using open book decompositions. It has been conjectured that these invariants agree where they are both defined. We prove this fact by defining yet another invariant of transverse knots, showing that this third invariant agrees with the two mentioned above. 57M27; 57R58

Khovanov homology and knot Floer homology for pointed links

A well-known conjecture states that for any

Naturality in sutured monopole and instanton homology

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Instanton Floer homology and contact structures

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On the complexity of torus knot recognition

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On the Spectral Sequence from Khovanov Homology to Heegaard Floer Homology

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Khovanov homology, open books, and tight contact structures

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