Narrow Results

Framed oriented $n$-tangle diagrams in the annulus, subject to the Homfly skein relations, are used to produce an algebra isomorphic to the affine Hecke algebras ${Ḣ}_n$ of type $A$. The use of closed curves and braids gives neat pictures for central elements in the algebras.

We define notions of pivotal and ribbon objects in a monoidal category. These constructions give pivotal or ribbon monoidal categories from a monoidal category which is not necessarily with duals. We apply this construction to the braided monoidal category of Yetter–Drinfeld modules over a Hopf algebra. This gives rise to the notion of ribbon Yetter–Drinfeld modules over a Hopf algebra, which form ribbon categories. This gives an invariant of tangles.

We solve a number of problems in quantum computing by applying genetic algorithms. We use the bitset class of C++ to represent any data type for genetic algorithms. Thus we have a flexible way to solve any optimization problem. The Bell-CHSH inequality and entanglement measures are studied using genetic algorithms. Entangled states form the backbone for teleportation. The C++ code is also provided.

Invariants of virtual 2n-tangles t are defined using an analog of the Temperley-Lieb algebra. The invariants yield information about the bracket polynomial of any link in which t embeds.

Extending and reproving a recent result of D. Krebes, we give obstructions to the embedding of a tangle in a link.

A classical theorem of R. H. Bing states that a closed connected 3-manifold M is homeomorphic to the 3-sphere if and only if every knot in M is contained in a 3-ball. We give a simple proof of this characterization based on the surgery presentation of 3-manifolds.

The system of unoriented tangle equations and is completely solved for the tangles U and as a function of where K₁ and K₂ are 4-plats, and and rational tangles such that |f₁g₂ - g₁f₂| > 1.

The system of unoriented tangle equations and is completely solved for the tangles U and as a function of where K₁ and K₂ are 4-plats, and and rational tangles such that |f₁g₂ - g₁f₂| > 1. As an application, it is completely determined when one 4-plat can be obtained from another 4-plat via a signed crossing change.

It is conjectured that a hyperbolic knot admits at most three Dehn surgeries which yield closed 3-manifolds containing incompressible tori. We show that there exist infinitely many hyperbolic knots which attain the conjectural maximum number. Interestingly, those surgeries correspond to consecutive integers.

We use a special kind of 2-dimensional extended Topological Quantum Field Theories (TQFTs), so-called open-closed TQFTs, in order to represent a refinement of Bar-Natan's universal geometric complex algebraically, and thereby extend Khovanov homology from links to arbitrary tangles. For every plane diagram of an oriented tangle, we construct a chain complex whose terms are modules of a suitable algebra A such that there is one action of A or A^op for every boundary point of the tangle. We give examples of such algebras A for which our tangle homology theory reduces to the link homology theories of Khovanov, Lee and Bar-Natan if it is evaluated for links. As a consequence of the Cardy condition, Khovanov's graded theory can only be extended to tangles if the underlying field has finite characteristic. Whenever the open-closed TQFT arises from a state-sum construction, we obtain honest planar algebra morphisms, and all composition properties of the universal geometric complex carry over to the algebraic complex. We give examples of state-sum open-closed TQFTs for which one can still determine both characteristic p Khovanov homology of links and Rasmussen's s-invariant.

A generalization of the Kauffman tangle algebra is given for Coxeter type D_n. The tangles involve a pole of order 2. The algebra is shown to be isomorphic to the Birman–Murakami–Wenzl algebra of the same type. This result extends the isomorphism between the two algebras in the classical case, which, in our set-up, occurs when the Coxeter type is A_{n - 1}. The proof involves a diagrammatic version of the Brauer algebra of type D_n of which the generalized Temperley–Lieb algebra of type D_n is a subalgebra.

In this article we extend evaluations of the Kauffman bracket on regular isotopy classes of knots and links to a variety of functors defined on the category of framed tangles. We show that many such functors exist, and that they correspond up to equivalence to bilinear forms on free, finitely-generated modules over commutative rings R.

We consider the problem of classification of links up to (2, 2)-moves. Our motivation comes from the theory of skein modules, more specifically from the skein module of S³ based on the deformation of (2, 2)-move. As it was proved in D-P-2, not every link can be reduced to a trivial link by (2, 2)-moves, for instance, the closure of (σ₁σ₂)⁶. In this paper, we classify 3-braids up to (2, 2)-moves and, we show how the Harikae–Nakanishi–Uchida conjecture can be modified to hold for closed 3-braids. As an important step in the classification we prove the conjecture for 2-algebraic links and classify (2, 2)-equivalence classes for links up to nine crossings. We also analyze an action of (2, 2)-move on Kei (involutive quandle) associated to a link diagram. We define Burnside Kei, Q(m, n), and ask for which values of m and n, is Q(m, n) finite. This question is motivated by classical Burnside problem.

In earlier work the Kauffman bracket polynomial was extended to an invariant of marked graphs, i.e. looped graphs whose vertices have been partitioned into two classes (marked and not marked). The marked-graph bracket polynomial is readily modified to handle graphs with weighted vertices. We present formulas that simplify the computation of this weighted bracket for graphs that contain twin vertices or are constructed using graph composition, and we show that graph composition corresponds to the construction of a link diagram from tangles.

We give a new construction of the one-variable Alexander polynomial of an oriented knot or link, and show that it generalizes to a vector valued invariant of oriented tangles.

In this paper, we define the notion of descending tangle diagrams and give a presentation of a link as a sum of two descending tangle diagrams. We also introduce a new invariant of links and find its basic properties.

Three disjoint rays in ℝ³ form Borromean rays provided their union is knotted, but the union of any two components is unknotted. We construct infinitely many Borromean rays, uncountably many of which are pairwise inequivalent. We obtain uncountably many Borromean hyperplanes.

Given a compact oriented 3-manifold M in S³ with boundary, an (M, 2n)-tangle is a 1-manifold with 2n boundary components properly embedded in M. We say that embeds in a link L in S³ if can be completed to L by a 1-manifold with 2n boundary components exterior to M. The link L is called a closure of . We define the Kauffman bracket ideal of to be the ideal of ℤ[A, A^-1] generated by the reduced Kauffman bracket polynomials of all closures of . If this ideal is non-trivial, then does not embed in the unknot. We give an algorithm for computing a finite list of generators for the Kauffman bracket ideal of any (S¹ × D², 2)-tangle, also called a genus-1 tangle, and give an example of a genus-1 tangle with non-trivial Kauffman bracket ideal. Furthermore, we show that if a single-component genus-1 tangle can be obtained as the partial closure of a (B³, 4)-tangle , then .

An n-string tangle is a three-dimensional ball with n-strings properly embedded in it. In the late 1980s, Ernst and Sumners introduced a tangle model for protein-DNA complexes. The protein is modeled by a three-dimensional ball and the protein-bound DNA is modeled by strings embedded inside the ball. Originally the tangle model was applied to proteins such as Tn3 resolvase which binds two DNA segments. This protein breaks and rejoins two DNA segments and can create knotted DNA. A 2-string tangle model can be used for this complex. More recently, Pathania, Jayaram and Harshey determined that the topological structure of DNA within a Mu protein complex consists of three DNA segments containing five crossings. Since Mu binds DNA sequences at three sites, this Mu protein-DNA complex can be modeled by a 3-string tangle. Darcy, Leucke and Vazquez analyzed Pathania et al.'s experimental results by using 3-string tangle analysis. There are protein-DNA complexes that involve four or more DNA sites. When a protein binds circular DNA at four sites, a protein-DNA complex can be modeled by a 4-string tangle with four loops outside of the tangle. We determine a biologically relevant 4-string tangle model. We also develop mathematics for solving 4-string tangle equations to determine the topology of DNA within a protein complex.

We show that every alternating link of two components and $12$ crossings can be reduced by $4$-moves to the trivial link or the Hopf link. It answers the question asked in one of the last papers by Slavik Jablan.