Publications (13)0 Total impact
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ABSTRACT: We produce the first examples of closed, tight contact 3-manifolds which
become overtwisted after performing admissible transverse surgeries. Along the
way, we clarify the relationship between admissible transverse surgery and
Legendrian surgery. We use this clarification to study a new invariant of
transverse knots - namely, the range of slopes on which admissible transverse
surgery preserves tightness - and to provide some new examples of knot types
which are not uniformly thick. Our examples also illuminate several interesting
new phenomena, including the existence of hyperbolic, universally tight contact
3-manifolds whose Heegaard Floer contact invariants vanish (and which are not
weakly fillable); and the existence of open books with arbitrarily high
fractional Dehn twist coefficients whose compatible contact structures are not
deformations of co-orientable taut foliations.
03/2012;
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ABSTRACT: 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.
12/2011;
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ABSTRACT: We iterate Manolescu's unoriented skein exact triangle in knot Floer homology
with coefficients in the fraction field of the group ring (Z/2Z)[Z]. The result
is a spectral sequence which converges to a stabilized version of delta-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.
05/2011;
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John A. Baldwin
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ABSTRACT: Suppose S is a compact surface with boundary, and let g be a diffeomorphism of S which fixes the boundary pointwise. We denote by (M_{S,g},\xi_{S,g})$ the contact 3-manifold compatible with the open book (S,g). In this article, we construct a Stein cobordism from the contact connected sum (M_{S,h},\xi_{S,h}) # (M_{S,g},\xi_{S,g}) to (M_{S,hg},\xi_{S,hg}), for any two boundary-fixing diffeomorphisms h and g. This cobordism accounts for the comultiplication map on Heegaard Floer homology discovered in an earlier paper by the author, and it illuminates several geometrically interesting monoids in the mapping class group of S. We derive some consequences for the fillability of contact manifolds obtained as cyclic branched covers of transverse knots. Comment: 12 pages, 5 figures
05/2010;
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ABSTRACT: In this note we observe that the no two of the three invariants defined for contact structures by Etnyre and Ozbagci -- that is, the support genus, binding number and support norm -- determine the third. Comment: 10 pages, 5 figures
10/2009;
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John A. Baldwin
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ABSTRACT: For a word w in the braid group on n-strands, we denote by T_w the corresponding transverse braid in the rotational symmetric tight contact structure on S^3. We exhibit a map on link Floer homology which sends the transverse invariant associated to T_{ws_i} to that associated to T_w, where s_i is one of the standard generators of B_n. This gives rise to a "comultiplication" map on link Floer homology. We use this to generate infinitely many new examples of prime topological link types which are not transversely simple. Comment: 16 pages, 10 figures
10/2009;
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John A. Baldwin
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ABSTRACT: If (S,h) is an open book with disconnected binding then we can form a new open book (S',h') by capping off one of the boundary components of S with a disk. We define a U-equivariant map on Heegaard Floer homology which sends c^+(S',h') to c^+(S,h), and we discuss various applications. In particular, we determine the support genera of almost all contact structures compatible with genus one, one boundary component open books. In addition, we compute the 3-dimensional invariant associated to any contact structure with non-vanishing contact invariant which is compatible with a genus one open book with periodic monodromy. Comment: 29 pages, 15 figures. Added section on support genera for genus one, one boundary component open books. Added result about gluing open books
01/2009;
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John A. Baldwin
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ABSTRACT: Ozsvath and Szabo show that there is a spectral sequence whose E^2 term is the reduced Khovanov homology of L, and which converges to the Heegaard Floer homology of the (orientation reversed) branched double cover of S^3 along L. We prove that the E^k term of this spectral sequence is an invariant of the link L for all k >= 2. If L is a transverse link in the standard tight contact structure on S^3, then we show that Plamenevskaya's transverse invariant psi(L) gives rise to a transverse invariant, psi^k(L), in the E^k term for each k >= 2. We use this fact to compute each term in the spectral sequences associated to the torus knots T(3,4) and T(3,5). Comment: Added 2 sections of background and section on independence of analytic data
09/2008;
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John A. Baldwin
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ABSTRACT: We compute the Heegaard Floer homology of any rational homology 3-sphere with an open book decomposition of the form (T,\phi), where T is a genus one surface with one boundary component. In addition, we compute the Heegaard Floer homology of any T^2-bundle over S^1 with first Betti number equal to one, 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-filllings 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.
05/2008;
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John A. Baldwin
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ABSTRACT: Suppose that S is a surface with boundary and that g and h are diffeomorphisms of S which restrict to the identity on the boundary. Let Y_g, Y_h, and Y_{hg} be the three-manifolds with open book decompositions given by (S,g), (S,h), and (S,hg), respectively. We show that the Ozsvath-Szabo contact invariant is natural under a comultiplication map on Heegaard Floer homology. It follows that if the contact invariants associated to the open books (S, g) and (S, h) are non-zero then the contact invariant associated to the open book (S, hg) is also non-zero. We extend this comultiplication to a map on HF^+, and as a result we obtain obstructions to the three-manifold Y_{hg} being an L-space. We also use this to find restrictions on contact structures which are compatible with planar open books.
03/2007;
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ABSTRACT: Using a combinatorial approach described in a recent paper of Manolescu, Ozsv\'ath, and Sarkar we compute the Heegaard-Floer knot homology of all knots with at most 12 crossings as well as the $\tau$ invariant for knots through 11 crossings. We review the basic construction of \cite{MOS}, giving two examples that can be worked out by hand, and explain some ideas we used to simplify the computation. We conclude with a discussion of knot Floer homology for small knots, closely examining the Kinoshita-Teraska knot $KT_{2,1}$ and its Conway mutant.
11/2006;
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John A. Baldwin
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ABSTRACT: Supose that $Y$ is a lens space with $|H_1(Y; \mathbb{Z})|$ prime, and $Y$ does not contain a genus one fibered knot. We show that $Y$ contains a knot whose exterior is a once-punctured torus bundle if and only if $Y$ is the result of $p/q$-surgery on the trefoil. This partially answers a question posed by Ken Baker in a paper in which he gives a complete classification of genus one fibered knots contained in lens spaces. Combining Baker's classification with Moser's characterization of lens space surgeries on the trefoil, we generate an infinite family of lens spaces which do not contain any knot whose exterior is a once-punctured torus bundle.
08/2006;
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John A. Baldwin
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ABSTRACT: We study contact structures compatible with genus one open book decompositions with one boundary component. Any monodromy for such an open book can be written as a product of Dehn twists around dual non-separating curves in the once-punctured torus. Given such a product, we supply an algorithm to determine whether the corresponding contact structure is tight or overtwisted. We rely on Ozsv{\'a}th-Szab{\'o} Heegaard Floer homology in our construction and, in particular, we completely identify the $L$-spaces with genus one, one boundary component, pseudo-Anosov open book decompositions. Lastly, we reveal a new infinite family of hyperbolic three-manifolds with no co-orientable taut foliations, extending the family discovered in \cite{RSS}.
05/2006;
