arXiv subjects
Publications and source records attributed to Maciej Borodzik.
We show that for any two knots $K_1$, $K_2$ in $\mathbb{R}^3$, there exist diagrams $D_1$ and $D_2$ that represent $K_1$ and $K_2$, respectively, such that the Khovanov--Rozansky triply graded homologies of $D_1$ and $D_2$ are isomorphic. The methods expand on Abel's paper [Abe17].
An involutive link is a link which is invariant under the standard rotation by 180 degrees in $S^3$. We establish an equivariant analogue of the work of Carter and Saito aimed at studying equivariant cobordisms between involutive links. This gives a set of $39$ equivariant movie moves that suffice to go between any two movie presentations of a pair of equivariantly isotopic cobordisms. Along the way, we give a singularity-theoretic proof of the equivariant Reidemeister theorem and study loops of equivariant Reidemeister moves. Our approach proceeds by analyzing codimension $2$ singularities of equivariant maps from $S^1$ to $\mathbb{R}^2$, as well as utilizing embedded equivariant Morse theory.
We show that for every genus $g \geq 0$, there exist quasipositive knots $K_0^g$ and $K_1^g$ such that there is a cobordism of genus $g=|g_4(K_1^g)-g_4(K_0^g)|$ between $K_0^g$ and $K_1^g$, but there is no ribbon cobordism of genus $g$ in either direction and thus no complex cobordism between these two knots. This gives a negative answer to a question posed by Feller in 2016.
We study 1-parameter families of Morse functions for manifolds with boundary. We list all degeneracies that may occur in generic 1-parameter families.
We introduce a refinement of Bar-Natan homology for involutive links, extending the work of Lobb-Watson and Sano. We construct a new suite of numerical invariants and derive bounds for the genus of equivariant cobordisms between strongly invertible knots. Our invariants show that the difference between the equivariant slice genus and isotopy-equivariant slice genus can be arbitrarily large, whereas previously these were not known to differ.
We show that link concordance implies link homotopy for immersions of codimension at least two. As a consequence, we prove that every link $\sqcup^r S^n \hookrightarrow S^{n+2}$ is link homotopically trivial for $n\geq 2$, that is, there is a {\em link} homotopy that (for each time parameter) maps distinct component $n$-spheres disjointly into $S^{n+2}$. In other words, beyond the classical dimension (of embedded circles in $S^3$) there are no `linking modulo knotting' phenomena in codimension two. To date, this was only known for $n=2$. In our proofs we follow, expand, and complete unpublished notes of the third author developing stratified Morse theory for generic immersions, where the $d$-th stratum is given by points that have $d$ preimages under the generic immersion. We discuss gradient like vector fields, their strata preserving flows, and Cerf theory. This generalizes the case of embeddings, as studied by Perron, Sharpe, and Rourke, and expanded by the first two authors. Two vital operations in Cerf theory are the rearrangement and cancellation of critical points. In the setting of stratified theory, there are additional rearrangement and cancellation obstructions arising from intersections of ascending and descending membranes for critical points of the Morse function restricted to various strata. We show that both additional obstructions vanish in codimension at least three, implying a smooth proof of Hudson's result that embedded concordance implies isotopy. In codimension at least two, we show that only the rearrangement obstruction vanishes and we introduce finger moves that eliminate the cancellation obstruction. This is done carefully and only at the expense of introducing new self-intersection points into the components of the immersion. Therefore, our moves keep distinct components disjoint and hence preserve the link homotopy class.
In 2015, Borodzik, Némethi and Ranicki proved that an interior critical point can be pushed to the boundary, where it splits into two boundary critical points. In this paper, we show that two critical points at the boundary can be, under specific assumptions, merged into a single critical point in the interior. That is, we reverse the original construction.
We construct hyperbolic L-space knots that are not concordant to any linear combination of algebraic knots.
This paper describes how to compute algorithmically certain twisted signature invariants of a knot $K$ using twisted Blanchfield forms. An illustration of the algorithm is implemented on $(2,q)$-torus knots. Additionally, using satellite formulas for these invariants, we also show how to obstruct the sliceness of certain iterated torus knots.
We define a link lattice complex for plumbed links, generalizing constructions of Ozsváth, Stipsicz and Szabó, and of Gorsky and Némethi. We prove that for all plumbed links in rational homology 3-spheres, the link lattice complex is homotopy equivalent to the link Floer complex as an $A_\infty$-module. Additionally, we prove that the link Floer complex of a plumbed L-space link is a free resolution of its homology. As a consequence, we give an algorithm to compute the link Floer complexes of plumbed L-space links, in particular of algebraic links, from their multivariable Alexander polynomial.
We study local equivalence of bounded complexes over a polynomial ring $R[w]$, where $R$ is a noetherian ring. We provide a homological algebra approach to the results, the variants of which have been proved in many places in the literature.
For an $m$-periodic link $L$, we show that the Khovanov-Rozansky $\mathfrak{sl}_N$-homology carries an action of the group $\mathbb{Z}_m$. As an example of applications, we prove an analog of the periodicity criterion of Borodzik--Politarczyk using $\mathfrak{sl}_N$-homology instead of Khovanov homology.
This is the first paper in a series of three devoted to studying twisted linking forms of knots and three-manifolds. Its function is to provide the algebraic foundations for the next two papers by describing how to define and calculate signature invariants associated to a linking form $M\times M\to\mathbb{F}(t)/\mathbb{F}[t^{\pm1}]$ for $\mathbb{F}=\mathbb{R},\mathbb{C}$, where $M$ is a torsion $\mathbb{F}[t^{\pm 1}]$-module. Along the way, we classify such linking forms up to isometry and Witt equivalence and study whether they can be represented by matrices.
This paper studies twisted signature invariants and twisted linking forms, with a view towards obstructions to knot concordance. Given a knot $K$ and a representation $ρ$ of the knot group, we define a twisted signature function $σ_{K,ρ} \colon S^1 \to \mathbb{Z}$. This invariant satisfies many of the same algebraic properties as the classical Levine-Tristram signature $σ_K$. When the representation is abelian, $σ_{K,ρ}$ recovers $σ_K$, while for appropriate metabelian representations, $σ_{K,ρ}$ is closely related to the Casson-Gordon invariants. Additionally, we prove satellite formulas for $σ_{K,ρ}$ and for twisted Blanchfield forms.
We show that if a closed $C^1$-smooth surface in a Riemannian manifold has bounded Kolasinski--Menger energy, then it can be triangulated with triangles whose number is bounded by the energy and the area. Each of the triangles is an image of a subset of a plane under a diffeomorphism whose distortion is bounded by $\sqrt{2}$.
We study possible configurations of singular points occuring on general algebraic curves in $\mathbb{C}P^2$ via Floer theory. To achieve this, we describe a general formula for the $H_{1}$-action on the knot Floer complex of the knotification of a link in $S^3$, in terms of natural actions on the link Floer complex of the original link. This result may be interest on its own.
Given an $m$-periodic link $L\subset S^3$, we show that the Khovanov spectrum $\mathcal{X}_L$ constructed by Lipshitz and Sarkar admits a homology group action. We relate the Borel cohomology of $\mathcal{X}_L$ to the equivariant Khovanov homology of $L$ constructed by the second author. The action of Steenrod algebra on the cohomology of $\mathcal{X}_L$ gives an extra structure of the periodic link. Another consequence of our construction is an alternative proof of the localization formula for Khovanov homology, obtained first by Stoffregen and Zhang. By applying Dwyer-Wilkerson theorem we express Khovanov homology of the quotient link in terms of equivariant Khovanov homology of the original link.
We show that under a precise condition on the single variable Alexander polynomial, the limit at $1$ of the Tristram--Levine signature of a link is determined by the linking matrix.