Related papers: Knot Theory and the Problem of Classification for …

In this paper the new model of plastic deformation is constructed. For classification of plastic statuses the theory of knot is used.

Data science offers a powerful tool to understand objects in multiple sciences. In this paper we utilize concept of data science, most notably topological data analysis, to extend our understanding of knot theory. This approach provides a…

This paper builds a novel bridge between algebraic coding theory and mathematical knot theory, with applications in both directions. We give methods to construct error-correcting codes starting from the colorings of a knot, describing…

Knot theory is the Mathematical study of knots. In this paper we have studied the Composition of two knots. Knot theory belongs to Mathematical field of Topology, where the topological concepts such as topological spaces, homeomorphisms,…

We point out the connection between mathematical knot theory and spin glass/search problem. In particular, we present a statistical mechanical formulation of the problem of computing a knot invariant; p-colorability problem, which provides…

In a previous paper (q-alg/9501022) we suggested some algorithms that could be useful in solving the problem of knot classification. Here we continue this discussion by answering questions raised in that paper and by commenting on practical…

In this paper we summarise the work discussed in Ref. [1] and [2] (q-alg/9505003), in which we introduced a method helpful in solving the problem of knot classification. We also present results obtained since then.

We report a method for describing plasticity in a broad class of amorphous materials. The method is based on nonlinear (geometric) deformation theory allowing the separation of the plastic deformation from the general deformation tensor.…

I present a summary of the recent progress made in field and string theory which has led to a reformulation of quantum-group polynomial invariants for knots and links into new polynomial invariants whose coefficients can be described in…

The three-dimensional elastic-plastic deformation is considered. The catastrophe theory underlies the construction of this process model. It was shown that the variety of stable states consists on elastic states and can be depicted as a…

We present an elementary introduction to one of the most important today knot theory approaches, which gives rise to a representation for a class of knot polynomials in terms of quantum groups. Historically, the approach was at the same…

The study of knots and links from a probabilistic viewpoint provides insight into the behavior of "typical" knots, and opens avenues for new constructions of knots and other topological objects with interesting properties. The knotting of…

Knot theory provides a powerful tool for the understanding of topological matters in biology, chemistry, and physics. Here knot theory is introduced to describe topological phases in the quantum spin system. Exactly solvable models with…

Quantum phases can be classified by topological invariants, which take on discrete values capturing global information about the quantum state. Over the past decades, these invariants have come to play a central role in describing matter,…

This paper is a survey of knot theory and invariants of knots and links from the point of view of categories of diagrams. The topics range from foundations of knot theory to virtual knot theory and topological quantum field theory.

We review recent developments in the theory of Thompson group representations related to knot theory.

A polynomial is presented that models a topological knot in a unique manner. It distinguishes all types of knots including the orientation and has a group theory interpretation. The topologies may be labeled via a number, which upon a base…

This paper is a very brief introduction to knot theory. It describes knot coloring by quandles, the fundamental group of a knot complement, and handle-decompositions of knot complements.

Using methods of descriptive theory it is shown that the classification problem for wild knots is strictly harder than that for countable structures.

Knotted line defects in continuous fields entrain a complex arrangement of the material sur- rounding them. Recent experimental realisations in optics, fluids and nematic liquid crystals make it important to fully characterise these…