Journal of Knot Theory and Its Ramifications

We construct functors from a certain algebraic category , defined by Hopf algebra generators and relations, to the category of vector spaces, based on spherical categories. The category is proposed by Habiro to be isomorphic to the cobordism category of once-punctured surfaces. If the proposal is proved valid, the result of this paper would imply a construction of a TQFT functor based on a spherical category.

We study equivalence classes of knots and links of 2 components modulo 4-move. We show that all knots up to 12 crossings and knots in the family 6* reduce by 4-moves to the trivial knot. We also prove that links of 2 components with 11 crossings, and links 6* a₁.a₂.a₃.a₄.a₅.a₆ such that a_i is a 2-algebraic tangle with no trivial components reduce to either the trivial link or to the Hopf link. For alternating links of 2-components with 12 we show that L reduces by 4-moves to either trivial link or to the Hopf link whenever L is different than 9*.2 : .2 : .2 (or its mirror image). We suggest the alternating link 9*.2 : .2 : .2 with 12 crossings as a potential example to answer the Problem 1.1(iii) in negative.

We introduce a recoupling theory for virtual braided trees. This recoupling theory can be utilized to incorporate swap gates into anyonic models of quantum computation.

We characterize planar diagrams which may be divided into n arc embeddings in terms of their chord diagrams, generalizing a result of Taniyama for the case n = 2. Two algorithms are provided, one which finds a minimal arc embedding (in quadradic time in the number of crossings), and one which constructs a minimal subdiagram having the same arc number as D.

In a recent work "Arc-presentation of links: Monotonic simplification", Dynnikov shows that each rectangular diagram of the unknot, composite link, or split link can be monotonically simplified into a trivial, composite, or split diagram, respectively. The following natural question arises: Is it always possible to simplify monotonically a rectangular diagram of a satellite knot or link into one where the satellite structure is seen? Here we give a negative answer to that question both for knot and link cases.

An example of a torus embedding that cannot be obtained from ordinary "thin" torus by methods of paper [2] is also constructed.

Explicit chain homotopy maps and chain maps for the Reidemeister moves of Khovanov homology are often useful for several proofs of the isotopy invariance of Khovanov homology. However, such maps are missing except for the first Reidemeister moves given by Viro. In this paper, such chain homotopy maps and chain maps are obtained explicitly for the second and third Reidemeister moves (Sec. 2). Some applications are given to show the usefulness of these maps (Sec. 3).

Given a graph, we construct homology groups whose Euler characteristic is the Penrose polynomial of the graph, evaluated at an integer. This work is motivated by Khovanov's work on the categorification of the Jones polynomial for knots, and the subsequent categorifications of the chromatic and Tutte polynomials for graphs.

Diagrams and Reidemeister moves for links in orientable Seifert manifolds are introduced. Using these diagrams, we compute the Kauffman bracket skein modules of prism manifolds with first homology of order 4. In particular, we show that these modules are free.

We prove that if G is an abelian group of odd order then there is an isomorphism from the second quandle homology to G ∧ G, where ∧ is the exterior product. In particular, for , k odd, we have . Nontrivial allows us to use 2-cocycles to construct new quandles from T(G), and to construct link invariants. Computation of is also the first, fundamental step in the direction of computing homology of Takasaki quandles in general.

Some of the fundamental issues in quantum theory related to quantum computing are reviewed and discussed. Particularly emphasized is the need to be diligent in what quantum theory predicts, and what it does not. The non-intuitive features of quantum theory, that are often associated with aspects of non-locality and that also arise in quantum computing and quantum information transfer, are described. Some discussion of topological quantum computing using space-time strings is presented, as well as general notions about quantum computing.

We investigate properties of the odd Khovanov homology, compare and contrast them with those of the original (even) Khovanov homology, and discuss applications of the odd Khovanov homology to other areas of knot theory and low-dimensional topology.

We prove that the Murasugi–Przytycki index of the link graph determines an upper bound for the number of reducing operations that can be performed on a link diagram to reduce the number of Seifert circles.