arXiv research

Study of equivariant movie moves for involutive links.

Study of equivariant ribbon concordance using Khovanov homology.

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 …

Proves an equivariant version of Heegaard Floer link surgery formula.

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. T…

Study of strongly invertible Legendrian links in contact 3-space.

Study reveals $\mathfrak{sl}_2$ action on link homology of specific torus links.

New homology for links in annulus discovered.

Link homology compared with geometric link invariants using Bott-Samelson varieties.

New $\mathfrak{sl}_2$ action on link homologies discovered.

Specialized knot theory theorems for strongly involutive links.

New invariants refine link homology, showing large genus differences.

Study signatures of torus links and their cores using Neumann's equivariant signatures and Hirzebruch's formula.

Sign equivariant networks improve model expressiveness for spectral geometric learning.

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 t…

Let $Δ$ be a trivial knot in the three-sphere. For every finite cyclic group $G$ of odd order, we construct a $G$-equivariant Khovanov homology with coefficients in the filed $\F_{2}$. This homology is an invariant of links up to isotopy in $(S^{3},Δ)$. Another interpretation is given using the categorification of the …

Clarifies structures in link homology theories using Frobenius extensions.

We give a purely combinatorial formula for evaluating closed decorated foams. Our evaluation gives an integral polynomial and is directly connected to an integral equivariant version of the $\mathfrak{sl}_N$ link homology categorifying the $\mathfrak{sl}_N$ link polynomial. We also provide connections to the equivarian…

New invariants derived from link homology for 4-manifolds.

Witt algebra acts on Khovanov-Rozansky homology of links.

We construct an equivariant colored sl(N)-homology for links, which generalizes both the colored sl(N)-homology defined by the author and the equivariant sl(N)-homology defined by Krasner. The construction is a straightforward generalization of that of the colored sl(N)-homology. The proof of invariance is based on a s…

Derives formulae for general permutation equivariant layers and presents a second order graph variational encoder.

This work extends knot homology theory to links, proving exact triangles and categorifying link signatures.

For every positive integer $n$ we construct a bigraded homology theory for links, such that the corresponding invariant of the unknot is closely related to the U(n)-equivariant cohomology ring of $\mathbb{CP}^{n-1}$; our construction specializes to the Khovanov-Rozansky $sl_n$-homology. We are motivated by the "univers…

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 …

Refines quantum annular homology using stable homotopy methods.

A new polynomial invariant for strongly involutive links.

We explain how rank two Frobenius extensions of commutative rings lead to link homology theories and discuss relations between these theories, Bar-Natan theories, equivariant cohomology and the Rasmussen invariant.

Let G be a compact Lie group and A(G) its Burnside Ring. For a compact smooth n-dimensional G-manifold X equipped with a generic G-invariant vector field v, we prove an equivariant analog of the Morse formula Ind^G(v) = \sum_{k = 0}^{n} (-1)^k χ^G(\d_k^+X) which takes its values in A(G). Here Ind^G(v) denotes the equiv…

Novel threefold partitioning of $L^2$-spinor space on cone links.

We compute the average Tristram---Levine signature of any graph link with positive weights in a three sphere, generalizing the results of Kirby and Melvin. The main tools are the Neumann's algorithm for computing the equivariant signatures of graph links and the Reciprocity Law for Dedekind sums.

We define an invariant of based transverse links, as a well-defined element inside the equivariant Heegaard Floer cohomology of its branched double cover, defined by Lipschitz, Hendricks, and Sarkar. We prove the naturality and functoriality of equivariant Heegaard Floer cohomology for branched double covers of $S^3$ a…

We will discuss the equivariant cohomology of a manifold endowed with the action of a Lie group. Localization formulae for equivariant integrals are explained by a vanishing theorem for equivariant cohomology with generalized coefficients. We then give applications to integration of characteristic classes on symplectic…

Proves representability of complex semigroup systems.

Program connects birational invariants with G-equivariant ones using Gromov-Witten theory.

We construct equivariant Khovanov spectra for periodic links, using the Burnside functor construction introduced by Lawson, Lipshitz, and Sarkar. By identifying the fixed-point sets, we obtain rank inequalities for odd and even Khovanov homologies, and their annular filtrations, for prime-periodic links in $S^3$.

The singular instanton Floer homology was defined by Kronheimer and Mrowka in connection with their proof that the Khovanov homology is an unknot detector. We study this theory for knots and two-component links using equivariant gauge theory on their double branched covers. We show that the special generator in the sin…

Let M be a closed oriented 3-manifold with first Betti number one. Its equivariant linking pairing may be seen as a two-dimensional cohomology class in an appropriate infinite cyclic covering of the space of ordered pairs of distinct points of M. We show how to define the equivariant cube Q(K) of this Blanchfield pairi…

Measures equivariance in vision models using Lie derivative.

Study equivariant isotopy in higher dimensions, finding exceptions.

Lobb observed in [arXiv:1103.1412] that each equivariant sl(N) Khovanov-Rozansky homology over C[a] admits a standard decomposition of a simple form. In the present paper, we derive a formula for the corresponding Lee-Gornik spectral sequence in terms of this decomposition. Based on this formula, we give a simple alter…

Let M be a closed oriented 3-manifold with first Betti number one. Its equivariant linking pairing may be seen as a two-dimensional cohomology class in an appropriate infinite cyclic covering of the configuration space of ordered pairs of distinct points of M. We show how to define the equivariant cube Q(M,K) of this B…

Study representation varieties of twisted Hopf links using combinatorial and Hodge theory.

We provide a finite dimensional categorification of the symmetric evaluation of $\mathfrak{sl}_N$-webs using foam technology. As an output we obtain a symmetric link homology theory categorifying the link invariant associated to symmetric powers of the standard representation of $\mathfrak{sl}_N$. In addition, the cons…

Proves non-solvability of concordance groups using Milnor invariants.

Invariants for colored links found, with topological protection of certain knots.

SMP model preserves proximity and permutation in graph neural networks.

Defines band maps in unoriented link Floer homology forming a skein exact triangle.