Related papers: Classical Knot Theory

We study the quandle counting invariant for a certain family of finite quandles with trivial orbit subquandles. We show how these invariants determine the linking number of classical two-component links up to sign.

The knot group is the fundamental group of a knot or link complement. A necessary and sufficient conditions for a group to be realized as the knot group of some link was provided. This result was shown using the closed braid method.…

In this paper we constructed new model of plastic deformation. The knot theory was used to classify the plastic state.

We discuss the theory of knots, and describe how knot invariants arise naturally in gravitational physics. The focus of this review is to delineate the relationship between knot theory and the loop representation of non-perturbative…

We resolve a case of the oriented knot complement conjecture by showing that knots in an orientable circle bundle $N$ over a genus $g \geq 2$ surface $S$ are determined by their complements. We apply this to the setting of canonical knots…

This paper discusses a generalization of virtual knot theory that we call multi-virtual knot theory. Multi-virtual knot theory uses a multiplicity of types of virtual crossings. As we will explain, this multiplicity is motivated by the way…

This paper starts a systematic description of colored knot polynomials, beginning from the first non-(anti)symmetric representation R=[2,1]. The project involves several steps: (i) parametrization of big families of knots a la…

We study algebraic tangles as fundamental components in knot theory, developing a systematic approach to classify and tabulate prime tangles using a novel canonical representation. The canonical representation enables us to distinguish…

The fundamental problem of knot theory is to know whether two knots are equivalent or not. As a tool to prove that two knots are different, mathematicians have developed various invariants. Knots invariants are just functions that can be…

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

This is survey about the classical knot concordance group, prepared for an upcoming handbook of knot theory. Topics include: the basic definitions of concordance; the theory of algebraic concordance as developed by Levine; the theory of…

We define a deformation of our earlier link homologies for fundamental representations of sl_m. The deformed homology of a link is isomorphic to the deformed homology of the disjoint union of its components. Moreover, there exists a…

Any knot group is the image of the group of a prime knot by a homomorphism that preserves peripheral structure. In fact, there are infinitely many such prime knots. A related partial order on knots is defined, and its properties are…

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 isomorphism type of the knot quandle introduced by Joyce is a complete invariant of tame knots. Whether two quandles are isomorphic is in practice difficult to determine; we show that this question is provably hard: isomorphism of…

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…

We have a knot quandle and a fundamental class as invariants for a surface-knot. These invariants can be defined for a classical knot in a similar way, and it is known that the pair of them is a complete invariant for classical knots. In…

The large size limit of matrix integrals with quartic potential may be used to count alternating links and tangles. The removal of redundancies amounts to renormalizations of the potential. This extends into two directions: higher genus and…

This paper is expository and is accessible to students. We define simple invariants of knots or links (linking number, Arf-Casson invariants and Alexander-Conway polynomials) motivated by interesting results whose statements are accessible…

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…