Related papers: Michel Kervaire work on knots in higher dimensions
In this paper we discuss the applications of knotoids to modelling knots in open curves and produce new knotoid invariants. We show how invariants of knotoids generally give rise to well-behaved measures of how much an open curve is…
We make use of the 3D nature of knots and links to find savings in computational complexity when computing knot invariants such as the linking number and, in general, most finite type invariants. These savings are achieved in comparison…
This is an expository article on diagrammatic representations of knots and links in various settings via braids.
We show that circular width is preserved under connected sum of knots for some cases.
We introduce and explore the relation between knot invariants and quiver representation theory, which follows from the identification of quiver quantum mechanics in D-brane systems representing knots. We identify various structural…
This paper visualizes a knot reduction algorithm
We obtain a quiver representation for a family of knots called double twist knots $K(p,-m)$. Particularly, we exploit the reverse engineering of Melvin-Morton-Rozansky(MMR) formalism to deduce the pattern of the charge matrix for these…
In hypercube approach to correlation functions in Chern-Simons theory (knot polynomials) the central role is played by the numbers of cycles, in which the link diagram is decomposed under different resolutions. Certain functions of these…
This is a brief overview of some applications of the ideas of abstract convexity to the upper semilattices of gauges in finite dimensions.
We define new notions of groups of virtual and welded knots (or links) and we study their relations with other invariants, in particular the Kauffman group of a virtual knot.
This is a review article on Lorenz knots.
We define a generalization of virtual links to arbitrary dimensions by extending the geometric definition due to Carter et al. We show that many homotopy type invariants for classical links extend to invariants of virtual links. We also…
We show that one can interweave an unknot into any non-alternating connected projection of a link so that the resulting augmented projection is alternating.
Biquandle brackets are a type of quantum enhancement of the biquandle counting invariant for oriented knots and links, defined by a set of skein relations with coefficients which are functions of biquandle colors at a crossing. In this…
The theory of operads (May, cyclic, modular, PROPs, etc) is extended to include higher dimensional phenomena, i.e. operations between operations, mimicking the algebraic structure on varieties of arbitrary dimensions, having marked…
The knots-quivers correspondence states that various characteristics of a knot are encoded in the corresponding quiver and the moduli space of its representations. However, this correspondence is not a bijection: more than one quiver may be…
We investigate relationships between bounds on the crossing number and the mosaic number of mosaic knots.
This paper introduces the concept of a Fourier knot. A Fourier knot is a knot that is represented by a parametrized curve in three dimensional space such that the coordinate functions are finite Fourier series in the parameter. The…
Hypercontractive inequalities are a useful tool in dealing with extremal questions in the geometry of high-dimensional discrete and continuous spaces. In this survey we trace a few connections between different manifestations of…
We explain new developments in classical knot theory in 3 and 4~dimensions, i.e. we study knots in 3-space, up to isotopy as well as up to concordance. In dimension~3 we give a geometric interpretation of the Kontsevich integral (joint with…