arXiv research

Whitehead doubles have matching meridional rank and bridge number.

Uniform convergence of isotopies implies ambient isotopy, aiding knot equivalence.

Origami can create complex knots, with minimum creases defining a new knot invariant.

Solenoids are ``inverse limits'' of the circle, and the classical knot theory is the theory of tame embeddings of the circle into the 3-space. We give some general study, including certain classification results, of tame embeddings of solenoids into the 3-space as the ``inverse limits'' of the tame embeddings of the ci…

Samuel J. Lomonaco Jr and Louis H. Kauffman conjectured that tame knot theory and knot mosaic theory are equivalent. We give a proof of the Lomonaco-Kauffman conjecture.

Constructs knots from 3-manifolds with specified geometric limits.

The paper explores orderability in link quandles and provides various results.

We show that every non-trivial tame knot or link in R^3 has a quadrisecant, i.e. four collinear points. The quadrisecant must be topologically non-trivial in a precise sense. As an application, we show that a nonsingular, algebraic surface in R^3 which is a knotted torus must have degree at least eight.

It is known that for every knotted curve in space, there is a line intersecting it in four places, a quadrisecant. Comparing the order of the four points along the line and knot we can distinguish three types of quadrisecants; the alternating ones have the most relevance for the geometry of a knot. In this paper we pro…

In this note, residual finiteness of quandles is defined and investigated. It is proved that free quandles and knot quandles of tame knots are residually finite and Hopfian. Residual finiteness of quandles arising from residually finite groups (conjugation, core and Alexander quandles) is established. Further, residual…

The distortion of a curve measures the maximum arc/chord length ratio. Gromov showed any closed curve has distortion at least pi/2 and asked about the distortion of knots. Here, we prove that any nontrivial tame knot has distortion at least 5pi/3; examples show that distortion under 7.16 suffices to build a trefoil kno…

In this paper, we prove that given two cubical links of dimension two in ${\mathbb R}^4$, they are isotopic if and only if one can pass from one to the other by a finite sequence of cubulated moves. These moves are analogous to the Reidemeister and Roseman moves for classical tame knots of dimension one and two, respec…

In 2008, Lomonaco and Kauffman introduced a knot mosaic system to define a quantum knot system. A quantum knot is used to describe a physical quantum system such as the topology or status of vortexing that occurs on a small scale can not see. Kuriya and Shehab proved that knot mosaic type is a complete invariant of tam…

New method uses mosaics to study wild knots.

Inspired by the paper on quantum knots and knot mosaics [23] and grid diagrams (or arc presentations), used extensively in the computations of Heegaard-Floer knot homology [2,3,7,24], we construct the more concise representation of knot mosaics and grid diagrams via mirror-curves. Tame knot theory is equivalent to knot…

In this paper, we prove than given two cubic knots $K_1$, $K_2$ in $\mathbb{R}^3$, they are isotopic if and only if one can pass from one to the other by a finite sequence of cubulated moves. These moves are analogous to the Reidemeister moves for classical tame knots. We use the fact that a cubic knot is determined by…

Let K be a tame knot with irreducible exterior M(K) in a closed, connected, orientable 3--manifold Sigma such that pi_1(Sigma) is cyclic. If infinity is not a strict boundary slope, then the diameter of the set of strict boundary slopes of K, denoted d_K, is a numerical invariant of K. We show that either (i) d_K >= 2 …

Given a tame knot K presented in the form of a knot diagram, we show that the problem of determining whether K is knotted is in the complexity class NP, assuming the generalized Riemann hypothesis (GRH). In other words, there exists a polynomial-length certificate that can be verified in polynomial time to prove that K…

We consider pairs (X,Y) where X is a compact, locally CAT(-1) space, and Y is a totally geodesic subspace. The inclusion induces an embedding of the boundaries at infinity of the universal covers; we focus on the case where these are spheres whose dimensions differ by 2. We show that if the embedding is tame, then it i…

This paper classifies links in 3D dynamical systems.

New tile types for knots and links reduce complexity.

We introduce the new class of submanifolds of co-Banach type in tame Fréchet manifolds and construct tame Fréchet submanifolds as inverse images of regular values of certain tame maps. Our method furnishes an easy way to construct tame Fréchet manifolds. The results presented are key ingredients in the construction of …

Paper characterizes tamed and weakened tamed four-manifolds using a new technique.

The tame flows are ``nice'' flows on ``nice'' spaces. The nice (tame) sets are the pfaffian sets introduced by Khovanski, and a flow $Φ: \mathbb{R}\times X\to X$ on pfaffian set $X$ is tame if the graph of $Φ$ is a pfaffian subset of $\mathbb{R}\times X\times X$. Any compact tame set admits plenty tame flows. We prove …

Deep learning models viewed through tame geometry for convergence guarantees.

New algebra invariant distinguishes Legendrian knots in convex surfaces.

Paper proposes a new method to stabilize noisy gradient algorithms.

The study confirms Gromov's speculation and provides bounds for taming symplectic structures.

Complete classification of links and spatial graphs with finite N-quandles.

The paper develops finite knot theory using ropelength-filtered Reidemeister graphs.

RELTA-SGLD stabilizes nonconvex SGLD updates with a lighter taming scheme.

We study random knots, which we define as a triple of random periodic functions (where a random function is a random trigonometric series, \[f(θ) = \sum_{k=1}^\infty a_k \cos (k θ) +b_k (\sin k θ),\] with $a_k, b_k$ are independent gaussian random variables with mean $0$ and variance $σ(k)^2$ - our results will depend …

We prove that the group STame($k^3$) of special tame automorphisms of the affine 3-space is not simple, over any base field of characteristic zero. Our proof is based on the study of the geometry of a 2-dimensional simply-connected simplicial complex C on which the tame automorphism group acts naturally. We prove that …

Novel method computes Kauffman bracket skein modules of small 3-manifolds.

A closed discrete subset $A\subset \mathbb{C}$ is called tame if $\mathbb{C}\setminus A$ is quasiconformally equivalent to $\mathbb{C}\setminus \mathbb{Z}$. By giving several criteria for $A$ to be tame, we shall show that $\mathbb{Z}+i\mathbb{Z}$ is not tame.

Positivity of intersections in 4-manifolds leads to taming symplectic structures.

In this paper, we give a precise and workable definition of a quantum knot system, the states of which are called quantum knots. This definition can be viewed as a blueprint for the construction of an actual physical quantum system. Moreover, this definition of a quantum knot system is intended to represent the "quantu…

The survey we are presenting is over 22 years old but it has still some ideas which where never published (except in Polish). This survey is the base of the third Chapter of my book: KNOTS: From combinatorics of knot diagrams to combinatorial topology based on knots, which is still in preparation (but compare http://ar…

Study Alexander polynomials of links in 3-torus.

Every countable compact subset of sphere is tame.

Stability of knots at low regularity, and symmetric critical knots for Möbius energy.

Four-dimensional, oriented Lie algebras $\mathfrak{g}$ which satisfy the tame-compatible question of Donaldson for all almost complex structures $J$ on $\mathfrak{g}$ are completely described. As a consequence, examples are given of (non-unimodular) four-dimensional Lie algebras with almost complex structures which are…

We give several versions of local and global inverse mapping theorem for tame non necessarily smooth, mappings. Here tame mapping means a mapping which is subanalytic or, more generally, definable in some o-minimal structure. Our sufficient conditions are formulated in terms of various properties (convexity, positivity…

Develops calculus for tamed Dirichlet spaces using measure theory.

In this paper we show that a complete and non-compact surface immersed in the Euclidean space with quadratic extrinsic area growth has finite total curvature provided the surface has tamed second fundamental form and admits total curvature. In such a case we obtain as well a generalized Chern-Osserman inequality. In th…

This paper shows how path spaces on two-level manifolds can be Hilbert manifold structures.

We classify Veech groups of tame non-compact flat surfaces. In particular we prove that all countable subgroups of $\mathbf{GL}_+(2,\R)$ avoiding the set of mappings of norm less than 1 appear as Veech groups of tame non-compact flat surfaces which are Loch Ness monsters. Conversely, a Veech group of any tame flat surf…

Symplectic forms taming complex structures on compact manifolds are strictly related to Hermitian metrics having the fundamental form $\partial \bar \partial$-closed, i.e. to strong Kähler with torsion (${\rm SKT}$) metrics. It is still an open problem to exhibit a compact example of a complex manifold having a tamed …