arXiv subjects

Wojciech Politarczyk

Publications and source records attributed to Wojciech Politarczyk.

15 recordsLinked to original sources

arXiv2025

Torres-type formulas for link signatures

We investigate the limits of the multivariable signature function $σ_L$ of a $μ$-component link $L$ as some variable tends to $1$ via two different approaches: a three-dimensional and a four-dimensional one. The first uses the definition of $σ_L$ by generalized Seifert surfaces and forms. The second relies on a new extension of $σ_L$ from its usual domain $(S^1\setminus\{1\})^μ$ to the full torus $\mathbb{T}^μ$ together with a Torres-type formula for $σ_L$, results which are of independent interest. Among several consequences, we obtain new estimates on the value of the Levine-Tristram signature of a link close to $1$.

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arXiv2024

Twisted Blanchfield pairings and twisted signatures III: Applications

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.

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arXiv2023

On the slice genus of generalized algebraic knots

We give examples of a linear combination of algebraic knots and their mirrors that are algebraically slice, but whose topological and smooth four-genus is two. Our examples generalize an example of non-slice algebraically slice linear combination of iterated torus knots obtained by Hedden, Kirk and Livingston. Our main tool is a genus bound from Casson--Gordon invariants and a cabling formula that allows us to compute effectively these invariants.

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arXiv2022

Khovanov-Rozansky $\mathfrak{sl}_N$-homology for periodic links

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.

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arXiv2022

Twisted Blanchfield pairings and twisted signatures I: Algebraic background

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.

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arXiv2022

A new polynomial criterion for periodic knots

The purpose of this paper is to present a new periodicity criterion. For that purpose, we study the HOMFLY-PT polynomial and the Kauffman polynomial of cables of periodic links. Furthermore, we exhibit a couple of examples for which our criterion is stronger than many previously know criteria, among which is the Khovanov homology criterion of Borodzik and the second author.

math.GT↗

arXiv2022

Twisted Blanchfield pairings and twisted signatures II: Relation to Casson-Gordon invariants

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.

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arXiv2021

Khovanov homotopy type, periodic links and localizations

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.

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arXiv2020

Non-slice linear combinations of iterated torus knots

In 1976, Rudolph asked whether algebraic knots are linearly independent in the knot concordance group. This paper uses twisted Blanchfield pairings to answer this question in the affirmative for new large families of algebraic knots.

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arXiv2017

Equivariant Khovanov Homology of Periodic Links

The purpose of this paper is to construct and study equivariant Khovanov homology - a version of Khovanov homology theory for periodic links. Since our construction works regardless of the characteristic of the coefficient ring it generalizes a previous construction by Chbili. We establish invariance under equivariant isotopies of links and study algebraic properties of integral and rational version of the homology theory. Moreover, we construct a skein spectral sequence converging to equivariant Khovanov homology and use this spectral sequence to compute, as an example, equivariant Khovanov homology of torus links $T(n,2)$.

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arXiv2017

Khovanov homology and periodic links

Based on the results of the second author, we define an equivariant version of Lee and Bar-Natan homology for periodic links and show that there exists an equivariant spectral sequence from the equivariant Khovanov homology to equivariant Lee homology. As a result we obtain new obstructions for a link to be periodic. These obstructions generalize previous results of Przytycki and of the second author.

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arXiv2015

Equivariant Jones Polynomials of periodic links

This paper continues the study of periodic links started in \cite{Politarczyk2}. It contains a study of the equivariant analogues of the Jones polynomial, which can be obtained from the equivariant Khovanov homology. In this paper we describe basic properties of such polynomials, show that they satisfy an analogue of the skein relation and develop a state-sum formula. The skein relation in the equivariant case is used to strengthen the periodicity criterion of Przytycki from \cite{Przytycki}. The state-sum formula is used to reproved the classical congruence of Murasugi from \cite{Murasugi1}.

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arXiv2013

Stable classification of 4-manifolds obtained by the surgery on loops

This paper is concerned with the problem of stable diffeomorphism classification of 4-manifolds obtained using the surgery on loops. The main theorem states that under the assumption that the normal 1-type of two 4-manifolds in question is the same, the only classifying invariant is the signature. In particular, in some cases, any two closed smooth 4-manifolds with a given fundamental group, obtained by the standard construction, are stably diffeomorphic.

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arXiv2013

4-manifolds, surgery on loops and geometric realization of Tietze transformations

In the paper \cite{wall_1}, C.T.C. Wall proved that two smooth closed simply connected 4-manifolds which are homeomorphic are in fact stably diffeomorphic. We prove a similar result which states that two smooth closed 4-manifolds satisfying certain properties are stably diffeomorphic if and only if their signatures agree. The manifolds in question are obtained by surgery on loops. The methods we use are modified surgery of Kreck \cite{kreck} and Kirby calculus.

math.GT↗

arXiv2011

Non-symplectic actions on complex projective spaces

We construct smooth actions of arbitrary compact Lie groups on complex projective spaces, such that the corresponding transformations arising from the group action do not preserve any symplectic structure on the complex projective space.

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