~ Research ~

My research interests are in the  topology of 3 and 4 dimensional manifolds and classical knot theory. My primary emphasis as a graduate student was on Seiberg-Witten gauge theory with a special focus on symplectic 4-manifolds. My thesis [1] explored the relation between Seiberg-Witten and Gromov-Witten invariants in the symplectic category.

I have since shifted my attention to Heegaard Floer theory. In joint work with Thomas Mark (University of Virginia), I have calculated the invariants for certain 3-manifolds which fiber over the circle [3,4], including the case of a surface times a circle [9]. These calculations are a necessary precursor for the understanding of Heegaard Floer homology on 4-manifolds. A first step in this direction is a quite general product formula for 4-manifolds [10]. 

More recently, in joint work with Swatee Naik (University of Nevada Reno) and Josh Greene (Princeton University) I have applied Heegaard Floer homology to question about order in the smooth knot concordance group. Swatee and I [7] have defined a new obstruction for a knot to be of finite (but specific) concordance order n. In collaboration with Josh [11] I have solved the slice-ribbon conjecture for 3-stranded pretzel knots P(pq,r) with p,q,r odd.


~Publications and preprints~

1.  "Grafting Seiberg-Witten monopoles", Algebr. Geom. Topol. 3 (2003), 155-185
Abstract: It is demonstrated that the operation of taking disjoint unions of J-holomorphic curves (and thus obtaining new J-holomorphic curves) has a Seiberg-Witten counterpart. The main theorem asserts that, given two solutions (Ai, ψ i), i=0,1 of the Seiberg-Witten equations for the spinc-structures Wi+= Ei + (Ei K-1) (with certain restrictions), there is a solution (A, ψ ) of the Seiberg-Witten equations for the spinc-structure W =  E + (E K-1) with E= E0E1, obtained by  "grafting" the two solutions (Ai,ψ i).
2. "Dolbeaultova kohomologija na simplektičnim 4-mnogostrukostima" (in Croatian), Proceedings of the conference held in Ljubljana in June 2003 to mark the 30-th anniversary of the joint topological seminar Zagreb-Ljubljana.
Abstrakt: Proširujemo definiciju Dolbeaultovih grupa kohomologije s Kählerovih na opće simplektične mnogostrukosti. Kao primjenu, postavljamo i potvrdno odgovaramo na pitanje postoje li simplektične forme ω1 i ω2 na mnogostrukosti X4 s istim kanoničkim klasama K1 = K2 ali sa svojstvom da ne postoji difeomorfizam f:X→X za koji vrijedi f*(ω1) = ω2. U završnom poglavlju dajemo konkretan opis takvih formi na 4-dimenzionalnom torusu T4. Nepostojanje gore navedenog difeomorfizma f dokazujemo izračunavanjem Dolbeaultovih grupa. 
3. "Symplectic surfaces and generic J-holomorphic structures on four-manifolds", Illinois J. Math 48 (2004), no. 2, 675-685
Abstract: It is a well known fact that every embedded symplectic surface ∑ in a symplectic four-manifold  (X4,ω) can be made J-holomorphic for some almost-complex structure J compatible with ω. In this paper we investigate when such a J can be chosen generically in the sense of Taubes. The main result states that if ∑ is a connected embedded symplectic curve in X with genus g>0 and square ∑2 >g-2, then ∑ can be perturbed by in any of its tubular neighborhoods to obtain a J-holomorphic curve ∑' for some generic J. As an application we give examples of smooth and non-empty Seiberg-Witten and Gromov-Witten moduli spaces whose associated invariants are zero. 
4. "Heegaard Floer homology of certain mapping tori" (joint with Thomas Mark), Algebr. Geom. Topol. 4 (2004), 685-719
Abstract: We calculate the Heegaard Floer homologies HF+(M,s) for mapping tori M associated to certain surface diffeomorphisms, where s is any spinc-structure on M whose first Chern class is non-torsion. Let γ and δ be a pair of geometrically dual nonseparating curves on a genus g Riemann surface ∑g, and let σ be a curve separating ∑g into components of genera 1 and g-1. Write tγ, tδ, and tσ for the right-handed Dehn twists about each of these curves. The examples we consider are the mapping tori of the diffeomorphisms (tγ) m (tδ)n for integers m,n and that of tσ and (tσ)-1.
5. "Heegaard Floer homology of mapping tori II" (joint with Thomas Mark), Geometry and topology of manifolds, 119-135, Fields Inst. Commun., 47, Amer. Math. Soc., Providence, RI.
Abstract: We extend the techniques in a previous paper to calculate the Heegaard Floer homology groups HF+(M, s) for fibered 3-manifolds M whose monodromy is a power of a Dehn twist about a genus 1 separating circle on a surface of genus g>1, where s is a nontorsion spinc-structure on M.
6. "Torsion in Heegaard Floer homology" (joint with Thomas Mark), Oberwolfach Rep. 35 (2006), 28-31.
Abstract: We study the Heegaard Floer homology groups of a genus g surface times a circle. We exhibit that their HF+ groups carry 2-torsion (for g>2) and 3-torsion (for g>4). These are the first known examples to contain any torsion in either HF+, HF-, HF∞ or HF-hat. 
7. "Order in the concordance group and Heegaard Floer homology" (joint with Swatee Naik), Geom. Topol. 11 (2007), 979-994.
Abstract: We use the Heegaard-Floer homology correction terms defined by Ozsváth and Szabó to formulate a new obstruction for a knot to be of finite order in the smooth concordance group. This obstruction bears a formal resemblance to that of Casson and Gordon but is sensitive to the difference between the smooth versus topological category. As an application we obtain new lower bounds for the concordance order of small crossing knots.

An accompanying Mathematica script can be downloaded here. A free Mathematica Notebook Reader is available through the Wolfram Research website.
8. "Heegaard Floer homology and knot concordance: A survey of recent results", Glas. Mat. Ser. III, 42, no. 1 (2007), 237-256.
Abstract: This article surveys some recent advances made in the understanding of the smooth knot concordance group Con. The focus is exclusively on those results which have been driven by Heegaard Floer homology. Three invariants are discussed: the knot concordance homomorphisms τ, δ:Con→Z and the correction terms of double branched covers of knots.
9. "On the Heegaard Floer homology of a surface times a circle" (joint with Thomas Mark), Adv. Math. 218, no. 3 (2008), 728-761. 
Abstract: We make a detailed study of the Heegaard Floer homology of the product of a closed surface ∑g of genus g with S1. We determine HF+(∑gxS1,s;C) completely in the case of c1(s) =0, which for g>2 was previously unknown. We show that the case of HF∞ is closely related to the cohomology of the total space of a certain circle bundle over the Jacobian torus of ∑g and furthermore that HF+(∑gxS1,s;Z) contains nontrivial 2-torsion whenever g>2 and c1(s)=0.This is the first examples known to the authors of torsion in the Z-coefficients Heegaard Floer homology. Our methods also give new information about the action of H1(∑gxS1;Z) on HF+(∑gxS1,s) c when c1(s) is nonzero.
10. "Product formulae for Ozsvath-Szabo 4-manifold invariants" (joint with Thomas Mark), Geom. Topol. 12 (2008), 1557–1651.
Abstract: We give formulae for the Ozsváth-Szabó invariants of 4-manifolds X obtained by fiber sum of two manifolds M1 and M2 along surfaces ∑1 and ∑2 having trivial normal bundle and genus g>0. These formulae follow from a general gluing result of the Ozsváth-Szabó invariants phrased in terms of a pairing of relative invariants of the pieces. The fiber sum formula follows from this theorem along with previous results of the authors on the Heegaard Floer homology of manifolds of the form ∑xS1. The product formulae lead quickly to calculations of the Ozsváth-Szabó invariants of various 4-manifolds; in all cases the results are in accord with the conjectured equivalence between Ozsváth-Szabó and Seiberg-Witten invariants. 
11. "The slice-ribbon conjecture for 3-stranded pretzel knots" (joint with Josh Greene), Preprint 2007 (submitted for publication). 
Abstract: We determine the smooth concordance order of the 3-stranded pretzel knots P(p,q,r) with p,q,r odd. We show that each one of finite order is, in fact, ribbon, thereby proving the slice-ribbon conjecture for this family of knots. As corollaries we give new proofs of results first obtained by Fintushel-Stern and Casson-Gordon.
12. "Rational Witt classes of pretzel knots" Preprint 2008.
Abstract: In his pioneering work from 1969, Jerry Levine introduced a complete set of invariants of algebraic concordance of knots. The evaluation of these invariants requires a factorization of the Alexander polynomial of the knot, and is therefore in practice often hard to realize. We thus propose the study of an alternative set of invariants of algebraic concordance - the rational Witt classes of knots. Though these are rather weaker invariants than those defined by Levine, they have the advantage of lending themselves to quite manageable computability. We illustrate this point by computing the rational Witt classes of all pretzel knots. We give many examples and provide applications to obstructing sliceness for pretzel knots. We also obtain explicit formulae for the determinants and signatures of all pretzel knots. This article is dedicated to Jerry Levine and his lasting mathematical legacy; on the occasion of the conference "Fifty years since Milnor and Fox" held at Brandeis University on June 2-5, 2008.