Narrow Results

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.

In the present paper, we construct a virtual knot invariant with values in the free infinite-dimensional module over Z[a, a^-1]. The restriction of this invariant to the set of classical knots coincides with the Jones–Kauffman polynomial. It distinguishes virtual knots stronger than the generalised Jones-Kauffman polynomial proposed in [Kau].

The Kauffman bracket skein modules, , have been calculated for A=±1 for all 3-manifolds M by relating them to the -character varieties. We extend this description to the case when A is a 4th root of 1 and M is either a surface×[0,1] or a rational homology sphere (or its submanifold).

The paper computes the noncommutative A-ideal of the figure-eight knot, a noncommutative generalization of the A-polynomial. It is shown that if a knot has the same A-ideal as the figure-eight knot, then all colored Kauffman brackets are the same as those of the figure eight knot.

We discuss the strong invariant of virtual links proposed in [23]. This invariant is obtained as a generalization of the Jones–Kauffman polynomial (generalized Kauffman's bracket) by adding to the sum some equivalence classes of curves in two-dimensional surfaces. Thus, the invariant is valued in the infinite-dimensional free module over Z[q,q^-1].

We prove that this invariant can be decomposed into finite type Vassiliev invariant of virtual links (in Kauffman's sense); thus we present new infinite series of Vassiliev invariants.

It is also proved that this invariant is strictly stronger than the Jones–Kauffman polynomial for virtual knots proposed by Kauffman. Some examples when the invariant can recognize virtual knots that can not be recognized by other invariants are given.

We consider a class of topological objects in the 3-sphere S³ which will be called n-punctured ball tangles. Using the Kauffman bracket at A = e^iπ/4, an invariant for a special type of n-punctured ball tangles is defined. The invariant F takes values in PM_2×2ⁿ(ℤ), that is the set of 2 × 2ⁿ matrices over ℤ modulo the scalar multiplication of ±1. This invariant leads to a generalization of a theorem of Krebes which gives a necessary condition for a given collection of tangles to be embedded in a link in S³ disjointly. We also address the question of whether the invariant F is surjective onto PM_2×2ⁿ(ℤ). We will show that the invariant F is surjective when n = 0. When n = 1, n-punctured ball tangles will also be called spherical tangles We show that det F(S) ≡ 0 or 1 mod 4 for every spherical tangle S. Thus, F is not surjective when n = 1.

The Jones polynomial of a knot in 3-space is a Laurent polynomial in q, with integer coefficients. Many people have pondered why this is so, and what a proper generalization of the Jones polynomial for knots in other closed 3-manifolds is. Our paper centers around this question. After reviewing several existing definitions of the Jones polynomial, we argue that the Jones polynomial is really an analytic function, in the sense of Habiro. Using this, we extend the holonomicity properties of the colored Jones function of a knot in 3-space to the case of a knot in an integer homology sphere, and we formulate an analogue of the AJ Conjecture. Our main tools are various integrality properties of topological quantum field theory invariants of links in 3-manifolds, manifested in Habiro's work on the colored Jones function.

We introduce a local algorithm for Khovanov homology computations — that is, we explain how it is possible to "cancel" terms in the Khovanov complex associated with a ("local") tangle, hence canceling the many associated "global" terms in one swoosh early on. This leads to a dramatic improvement in computational efficiency. Thus our program can rapidly compute certain Khovanov homology groups that otherwise would have taken centuries to evaluate.

We derive a formula expanding the bracket with respect to a natural deformation parameter. The expansion is in terms of a two-variable polynomial algebra of diagram resolutions generated by basic operations involving the Goldman bracket. A functorial characterization of this algebra is given. Differentiability properties of the star product underlying the Kauffman bracket are discussed.

The celebrated Thistlethwaite theorem relates the Jones polynomial of a link with the Tutte polynomial of the corresponding planar graph. We give a generalization of this theorem to virtual links. In this case, the graph will be embedded into a (higher genus) surface. For such graphs we use the generalization of the Tutte polynomial discovered by Bollobás and Riordan.

The present paper is an introduction to a combinatorial theory arising as a natural generalization of classical and virtual knot theory. There is a way to encode links by a class of "realizable" graphs. When passing to generic graphs with the same equivalence relations we get "graph-links". On one hand graph-links generalize the notion of virtual link, on the other hand they do not detect link mutations. We define the Jones polynomial for graph-links and prove its invariance. We also prove some a generalization of the Kauffman–Murasugi–Thistlethwaite theorem on "minimal diagrams" for graph-links.

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.

A knot diagram has an associated looped interlacement graph, obtained from the intersection graph of the Gauss diagram by attaching loops to the vertices that correspond to negative crossings. This construction suggests an extension of the Kauffman bracket to an invariant of looped graphs, and an extension of Reidemeister equivalence to an equivalence relation on looped graphs. The graph bracket polynomial can be defined recursively using the same pivot and local complementation operations used to define the interlace polynomial, and it gives rise to a graph Jones polynomial V_G(t) that is invariant under the graph Reidemeister moves.

Let D be an oriented classical or virtual link diagram with directed universe . Let C denote a set of directed Euler circuits, one in each connected component of U. There is then an associated looped interlacement graph whose construction involves very little geometric information about the way D is drawn in the plane; consequently is different from other combinatorial structures associated with classical link diagrams, like the checkerboard graph, which can be difficult to extend to arbitrary virtual links. is determined by three things: the structure of as a 2-in, 2-out digraph, the distinction between crossings that make a positive contribution to the writhe and those that make a negative contribution, and the relationship between C and the directed circuits in arising from the link components; this relationship is indicated by marking the vertices where C does not follow the incident link component(s). We introduce a bracket polynomial for arbitrary marked graphs, defined using either a formula involving matrix nullities or a recursion involving the local complement and pivot operations; the marked-graph bracket of is the same as the Kauffman bracket of D. This provides a unified combinatorial description of the Jones polynomial that applies seamlessly to both classical and non-classical virtual links.

Based on the Kauffman bracket at A = e^iπ/4, we define an invariant for a special type of n-punctured ball tangles. The invariant Fⁿ takes values in the set PM_{2 × 2ⁿ}(ℤ) of 2 × 2ⁿ matrices over ℤ modulo the scalar multiplication of ±1. We provide the formula to compute the invariant of the k₁ + ⋯ + k_n-punctured ball tangle composed of given n, k₁, …, k_n-punctured ball tangles. Also, we define the horizontal and the vertical connect sums of punctured ball tangles and provide the formulae for their invariants from those of given punctured ball tangles. In addition, we introduce the elementary operations on the class ST of 1-punctured ball tangles, called spherical tangles. The elementary operations on ST induce the operations on PM_{2 × 2}(ℤ), also called the elementary operations. We show that the group generated by the elementary operations on PM_{2 × 2}(ℤ) is isomorphic to a Coxeter group.

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.

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.

Diagrams and Reidemeister moves for links in a twisted S¹-bundle over an unorientable surface are introduced. Using these diagrams, we compute the Kauffman Bracket Skein Module (KBSM) of ℝP³♯ℝP³. In particular, we show that it has torsion. We also present a new computation of the KBSM of S¹ × S² and the lens spaces L(p, 1).

In earlier work we introduced the graph bracket polynomial of graphs with marked vertices, motivated by the fact that the Kauffman bracket of a link diagram D is determined by a looped, marked version of the interlacement graph associated to a directed Euler system of the universe graph of D. Here we extend the graph bracket to graphs whose vertices may carry different kinds of marks, and we show how multiply marked graphs encode interlacement with respect to arbitrary (undirected) Euler systems. The extended machinery brings together the earlier version and the graph-links of Ilyutko and Manturov [J. Knot Theory Ramifications18 (2009) 791–823]. The greater flexibility of the extended bracket also allows for a recursive description much simpler than that of the earlier version.

This paper bounds the computational cost of computing the Kauffman bracket of a link in terms of the crossing number of that link. Specifically, it is shown that the image of a tangle with g boundary points and n crossings in the Kauffman bracket skein module is a linear combination of O(2^g) basis elements, with each coefficient a polynomial with at most n non-zero terms, each with integer coefficients, and that the link can be built one crossing at a time as a sequence of tangles with maximum number of boundary points bounded by for some C. From this it follows that the computation of the Kauffman bracket of the link takes time and memory a polynomial in n times .