Journal of Knot Theory and Its Ramifications 

About: Journal of Knot Theory and Its Ramifications is an academic journal published by World Scientific. The journal publishes majorly in the area(s): Knot theory & Knot (unit). It has an ISSN identifier of 0218-2165. Over the lifetime, 2485 publications have been published receiving 28265 citations.

Racks and links in codimension two

Abstract: A rack, which is the algebraic distillation of two of the Reidemeister moves, is a set with a binary operation such that right multiplication is an automorphism. Any codimension two link has a fundamental rack which contains more information than the fundamental group. Racks provide an elegant and complete algebraic framework in which to study links and knots in 3–manifolds, and also for the 3–manifolds themselves. Racks have been studied by several previous authors and have been called a variety of names. In this first paper of a series we consolidate the algebra of racks and show that the fundamental rack is a complete invariant for irreducible framed links in a 3–manifold and for the 3–manifold itself. We give some examples of computable link invariants derived from the fundamental rack and explain the connection of the theory of racks with that of braids.

Abstract link diagrams and virtual knots

Abstract: The notion of an abstract link diagram is re-introduced with a relationship with Kauffman's virtual knot theory. It is prove that there is a bijection from the equivalence classes of virtual link diagrams to those of abstract link diagrams. Using abstract link diagrams, we have a geometric interpretation of the group and the quandle of a virtual knot. A generalization to higher dimensional cases is introduced, and the state-sum invariants are treated.

Temperley-lieb algebras for non-planar statistical mechanics — the partition algebra construction

Abstract: We give the definition of the Partition Algebra Pn(Q). This is a new generalisation of the Temperley–Lieb algebra for Q-state n-site Potts models, underpinning their transfer matrix formulation on arbitrary transverse lattices. In Pn(Q) subalgebras appropriate for building the transfer matrices for all transverse lattice shapes (e.g. cubic) occur. For the Partition algebra manifests either a semi-simple generic structure or is one of a discrete set of exceptional cases. We determine the Q-generic and Q-independent structure and representation theory. In all cases (except Q = 0) simple modules are indexed by the integers j ≤ n and by the partitions λ ˫ j. Physically they may be associated, at least for sufficiently small j, to 2j 'spin' correlation functions. We exhibit a subalgebra isomorphic to the Brauer algebra.

Stable equivalence of knots on surfaces and virtual knot cobordisms

Abstract: We introduce an equivalence relation, called stable equivalence, on knot diagrams and closed generically immersed curves on surfaces. We give bijections between the set of abstract knots, the set of virtual knots, and the set of the stable equivalence classes of knot diagrams on surfaces. Using these bijections, we define concordance and link homology for virtual links. As an application, it is shown that Kauffman's example of a virtual knot diagram is not equivalent to a classical knot diagram.