Maciej Borodzik

HEEGAARD FLOER HOMOLOGY AND RATIONAL CUSPIDAL CURVES

We apply the methods of Heegaard Floer homology to identify topological properties of complex curves in the complex projective plane. As one application, we resolve an open conjecture that constrains the Alexander polynomial of the link of the singular point of the curve in the case that there is exactly one singular point, having connected link, and the curve is of genus 0. Generalizations apply in the case of multiple singular points.

Semigroups,

We apply Heegaard Floer homology to study deformations of singularities of plane algebraic curves. Our main result provides an obstruction to the existence of a deformation between two singularities. Generalizations include the case of multiple singularities. The obstruction is formulated in terms of a semicontinuity property for semigroups associated to the singularities.

d

Given a knot K we introduce a new invariant coming from the Blanchfield pairing and we show that it gives a lower bound on the unknotting number of K . This lower bound subsumes the lower bounds given by the Levine‐Tristram signatures, by the Nakanishi index and it also subsumes the Lickorish obstruction to the unknotting number being equal to one. Our approach in particular allows us to show for 25 knots with up to 12 crossings that their unknotting number is at least three, most of which are very difficult to treat otherwise. 57M27

-invariants and deformations of cuspidal singular points of plane curves

We develop Morse theory for manifolds with boundary. Besides standard and expected facts like the handle cancellation theorem and the Morse lemma for manifolds with boundary, we prove that, under a topological assumption, a critical point in the interior of a Morse function can be moved to the boundary, where it splits into a pair of boundary critical points. As an application, we prove that every cobordism of manifolds with boundary splits as a union of left product cobordisms and right product cobordisms.

The unknotting number and classical invariants, I

In this paper, we use topological methods to study various semicontinuity properties of the local spectrum of singular points of algebraic plane curves and spectrum at infinity of polynomial maps in two variables. Using the Seifert form, the Tristram‐Levine signatures of links, and the associated Murasugi-type inequalities, we reprove (in a slightly weaker form) a result obtained by Steenbrink and Varchenko on semicontinuity of the spectrum of singular points

Morse theory for manifolds with boundary

We use invariants of Hendricks and Manolescu coming from involutive Heegaard Floer theory to find constraints on possible configurations of singular points of a rational cuspidal curve of odd degree in the projective plane. We show that the results do not carry over to rational cuspidal curves of even degree.

Spectrum of plane curves via knot theory

Suppose C is a singular curve in CP^2 and it is topologically an embedded surface of genus g; such curves are called cuspidal. The singularities of C are cones on knots K_i. We apply Heegaard Floer theory to find new constraints on the sets of knots {K_i} that can arise as the links of singularities of cuspidal curves. We combine algebro-geometric constraints with ours to solve the existence problem for curves with genus one, d>33, that possess exactly one singularity which has exactly one Puiseux pair (p;q). The realized triples (p,d,q) are expressed as successive even terms in the Fibonacci sequence.

Involutive Heegaard Floer homology and rational cuspidal curves: RATIONAL CUSPIDAL CURVES

We use a knot invariant, namely the Tristram--Levine signature to study deformations of singular points of plane curves. We find a bound on the sum of M numbers over all singularities of a generic fiber in terms of the M number of the singularity at the central fiber and some topological data.

Plane algebraic curves of arbitrary genus via Heegaard Floer homology

In [BF12] the authors associated to a knot K ⊂ S an invariant nR(K) which is defined using the Blanchfield form and which gives a lower bound on the unknotting number. In this paper we express nR(K) in terms of Levine–Tristram signatures and nullities of K. In the proof we also show that the Blanchfield form for any knot K is diagonalizable over R[t].

Deformations of singularities of plane curves: Topological approach

We use purely topological methods to prove the semicontinuity of the mod 2 spectrum of local isolated hypersurface singularities in