arXiv research

Novel groups exhibit contradictory behaviors with respect to Burnside laws.

Let $n$ be a positive integer. M. K. Dabkowski and J. H. Przytycki introduced the $n$th Burnside group of links which is preserved by $n$-moves, and proved that for any odd prime $p$ there exist links which are not equivalent to trivial links up to $p$-moves by using their $p$th Burnside groups. This gives counterexamp…

A new presentation of a quotient of braid groups leads to a new type of Burnside group.

Extends quantum annular homology to infinite sets.

New Euler characteristic and Burnside group defined for definable groupoids.

Yasutaka Nakanishi asked in 1981 whether a 3-move is an unknotting operation. In Kirby's problem list, this question is called `The Montesinos-Nakanishi 3-move conjecture'. We define the n-th Burnside group of a link and use the 3rd Burnside group to answer Nakanishi's question; ie, we show that some links cannot be re…

Refines Khovanov homology using signed Burnside categories.

In a pair of recent papers (one to appear and one forthcoming), the author develops a general version of small cancellation theory applicable in higher dimensions, and then applies this theory to the Burnside groups of sufficiently large exponent. The present article gives a brief introduction to the methods and techni…

This paper is motivated by a general question: for which values of k and n is the universal Burnside kei of k generators and Kei "exponent" n, $\bar Q(k,n)$, finite? It is known (starting from the work of M. Takasaki (1942)) that $\bar Q(2,n)$ is isomorphic to the dihedral quandle Z_n and $\bar Q(3,3)$ is isomorphic to…

Using quantum representations of mapping class groups we prove that profinite completions of Burnside-type surface group quotients are not virtually prosolvable, in general. Further, we construct infinitely many finite simple characteristic quotients of surface groups.

We define and study the Burnside quotient Green ring of a Mackey functor. Some refinements of Dress induction theory are presented, together with applications to computation results for $K$-theory and $L$-theory of finite and infinite groups.

The study of periodic subgroups in homeomorphism groups of manifolds.

We define for arbitrary modules over a finite von Neumann algebra $\cala$ a dimension taking values in $[0,\infty]$ which extends the classical notion of von Neumann dimension for finitely generated projective $\cala$-modules and inherits all its useful properties such as additivity, cofinality and continuity. This all…

Functor decomposes Khovanov spectra for non-alternating diagrams.

In Classical Knot Theory and in the new Theory of Quantum Invariants substantial effort was directed toward the search for unknotting moves on links. We solve, in this note, several classical problems concerning unknotting moves. Our approach uses a new concept, Burnside groups of links, which establishes unexpected re…

There are (at least) two different approaches to define equivariant analogue of the Euler charateristic for a space with a finite group action. The first one defines it as an element of the Burnside ring of the group. The second approach emerged from physics and includes the orbifold Euler characteristic and its higher…

We classify 3-braids up to (2,2)-move equivalence and in particular we show how to adjust the Harikae-Nakanishi-Uchida conjecture so it holds for closed 3-braids. As important steps to classify 3-braids up to (2,2)-move equivalence we prove the conjecture for 2-algebraic links and classify (2,2)-equivalence classes for…

We discuss several open problems in classical Knot Theory and we develop techniques that allow us to study them: Lagrangian tangles, skein modules and Burnside groups.

Researchers found all embeddings of Kuratowski graphs on a double torus.

The purpose of this expository paper is to present new directions in the classical Nielsen-Reidemeister fixed point theory. We describe twisted Burnside-Frobenius theorem, groups with $R_\infty$ \emph{property} and a connection between Nielsen fixed point theory and symplectic Floer homology.

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 $S^3$.

In this note we present a combinatorial link invariant that underlies some recent stable homotopy refinements of Khovanov homology of links. The invariant takes the form of a functor between two combinatorial 2-categories, modulo a notion of stable equivalence. We also develop some general properties of such functors.

The purpose of the present paper is to prove for finitely generated groups of type I the following conjecture of A.Fel'shtyn and R.Hill, which is a generalization of the classical Burnside theorem. Let G be a countable discrete group, f one of its automorphisms, R(f) the number of f-conjugacy classes, and S(f)=# Fix (f…

In a previous paper, the authors defined an equivariant version of the so-called Saito duality between the monodromy zeta functions as a sort of Fourier transform between the Burnside rings of an abelian group and of its group of characters. Here a so-called enhanced Burnside ring $\widehat{B}(G)$ of a finite group $G$…

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…

We describe stable cup-i products on the cochain complex with $F^2$ coefficients of any augmented semi-simplicial object in the Burnside category. An example of such an object is the Khovanov functor of Lawson, Lipshitz and Sarkar. Thus we obtain explicit formulas for cohomology operations on the Khovanov homology of a…

Study shows equivariant Khovanov homotopy types are equivalent.

In this paper, we give a new construction of a Khovanov homotopy type. We show that this construction gives a space stably homotopy equivalent to the Khovanov homotopy types constructed in [LS14a] and [HKK] and, as a corollary, that those two constructions give equivalent spaces. We show that the construction behaves w…

The paper classifies Kleinian groups with Hausdorff dimension less than 1.

We construct a stable homotopy refinement of quantum annular homology, a link homology theory introduced by Beliakova, Putyra and Wehrli. For each $r\geq 2$ we associate to an annular link $L$ a naive $\mathbb{Z}/r\mathbb{Z}$-equivariant spectrum whose cohomology is isomorphic to the quantum annular homology of $L$ as …

We study a coarse homology theory with prescribed growth conditions. For a finitely generated group G with the word length metric this homology theory turns out to be related to amenability of G. We characterize vanishing of a certain fundamental class in our homology in terms of an isoperimetric inequality on G and sh…

In this article we extend the classical definitions of equivariant cohomotopy theory to the setting of proper actions of Lie groups. We combine methods originally developed in the analysis of nonlinear differential equations, mainly in connection with Leray-Schauder theory, and on the other hand from developments of eq…

Study on realizing subgroup twists in 3-manifolds.

New operations match Steenrod squares on Khovanov homology.

This work explains scaling laws as redundancy laws in deep learning.

Study on RL on volatility surfaces, proving no free lunch for law-seeking methods.

A theorem of Farb and Handel asserts that for $N\ge 4$, the natural inclusion from $\mathrm{Out}(F_N)$ into its abstract commensurator is an isomorphism. We give a new proof of their result, which enables us to generalize it to the case where $N=3$. More generally, we give sufficient conditions on a subgroup $Γ$ of $\m…

New 2-representations link spectral enhancements in link homology.

A new scaling law predicts optimal batch size for training models.

Space exploration technology advances exponentially, consistent with Moore's and Wright's laws.

We start a systematic analysis of links up to 5-move equivalence. Our motivation is to develop tools which later can be used to study skein modules based on the skein relation being deformation of a 5-move (in an analogous way as the Kauffman skein module is a deformation of a 2-move, i.e. a crossing change). Our main …

This work analyzes neural scaling laws using power-law data spectra and derives analytical expressions for generalization error.

An Atlas model is a rank-based system of continuous semimartingales for which the steady-state values of the processes follow a power law, or Pareto distribution. For a power law, the log-log plot of these steady-state values versus rank is a straight line. Zipf's law is a power law for which the slope of this line is …

Large models follow power laws in performance with dataset size or parameters.

Conservation law for weakly harmonic mappings in high dimensions.

Survey on conservation laws for geometric PDEs.

Unified theory for neural scaling laws in hierarchically compositional data.

Dynamic risk measures follow law invariance principles over time.