Related papers: Michel Kervaire work on knots in higher dimensions
Mosaic diagrams for knots were first introduced in 2008 by Lomanoco and Kauffman for the purpose of building a quantum knot system. Since then, many others have explored the structure of these knot mosaic diagrams, as they are interesting…
The Schur functions play a crucial role in the modern description of HOMFLY polynomials for knots and of topological vertices in DIM-based network theories, which could merge into a unified theory still to be developed. The Macdonald…
For any ring we propose the construction of a cover which increases the finitistic dimension on one side and decreases the finitistic dimension to zero on the opposite side. This complements recent work of Cummings.
In the present paper, we describe new approaches for constructing virtual knot invariants. The main background of this paper comes from formulating and bringing together the ideas of biquandle (Kauffman and Radford) the virtual quandle…
Two natural generalizations of knot theory are the study of spatially embedded graphs, and Kauffman's theory of virtual knots. In this paper we combine these approaches to begin the study of virtual spatial graphs.
This paper introduces a new approach to finding knots and links with hidden symmetries using "hidden extensions", a class of hidden symmetries defined here. We exhibit a family of tangle complements in the ball whose boundaries have…
In this paper, we give a geometric interpretation of virtual knotoids as arcs in thickened surfaces. Then we show that virtual knotoid theory is a generalization of classical knotoid theory. This gives a proof of a conjecture of Kauffman…
The purpose of this article is to present the theory of higher order connections on vector bundles from a viewpoint inspired by projective differential geometry.
We give a simple analytic criterion which characterizes linearizable 1-codimensional webs. Then we give an invariant geometrical interpretation of it, in term of projective connection. We explain then how our approach allows to study…
After Dirac introduced the monopole, topological objects have played increasingly important roles in physics. In this review we discuss the role of the knot, the most sophisticated topological object in physics, and related topological…
We elaborate on the spin projection operators in three dimensions and use them to derive a new representation for the linearised higher-spin Cotton tensors.
Virtual quandles with two operations are discussed in the article. Certain knot invariant is constructed and used to distinguish two long virtual knots.
In this paper, we introduce invariants of virtual knotoids based on biquandles and biquandle virtual brackets. We show that one of these invariants, namely biquandle virtual bracket matrix, is a proper enhancement of the other invariants…
Given a closed, oriented, connected 3-manifold, M, we define higher-order linking forms on the higher-order Alexander modules of M. These higher-order linking forms generalize similar linking forms for knots previously studied by the…
This paper formulates a generalization of our work on quantum knots to explain how to make quantum versions of algebraic, combinatorial and topological structures. We include a description of previous work on the construction of Hilbert…
This paper extends the construction of invariants for virtual knots to virtual long knots and introduces two new invariant modules of virtual long knots. Several interesting features are described that distinguish virtual long knots from…
The relation between open topological strings and representation theory of symmetric quivers is explored beyond the original setting of the knot-quiver correspondence. Multiple cover generalizations of the skein relation for boundaries of…
We analyze a variety of Weyl invariant dynamical problems in three dimensions.
We study surface knots in 4-space by using generic planar projections. These projections have fold points and cusps as their singularities and the image of the singular point set divides the plane into several regions. The width (or the…
Different possibilities of a detection of signals from extra space dimensions at high energy colliders are reviewed.