Narrow Results

We generalize Kauffman’s famous formula defining the Jones polynomial of an oriented link in $3$-space from his bracket and the writhe of an oriented diagram [L. Kauffman, State models and the Jones polynomial, Topology26(3) (1987) 395–407]. Our generalization is an epimorphism between skein modules of tangles in compact connected oriented $3$-manifolds with markings in the boundary. Besides the usual Jones polynomial of oriented tangles we will consider graded quotients of the bracket skein module and Przytycki’s $q$-analog of the first homology group of a $3$-manifold [J. Przytycki, A $q$-analogue of the first homology group of a $3$-manifold, in Contemporary Mathematics, Vol. 214 (American Mathematical Society, 1998), pp. 135–144]. In certain cases, e.g., for links in submanifolds of rational homology $3$-spheres, we will be able to define an epimorphism from the Jones module onto the Kauffman bracket module. For the general case we define suitably graded quotients of the bracket module, which are graded by homology. The kernels define new skein modules measuring the difference between Jones and bracket skein modules. We also discuss gluing in this setting.

We introduce a new skein module for three manifolds based on properly embedded surfaces and their relations introduced by Bar-Natan in [3], and modified by Khovanov [6]. We compute the structure of the modules for some manifolds, including Seifert fibered manifolds.

In this paper, we study the variation of the Turaev–Viro invariants for $3$-manifolds with toroidal boundary under the operation of attaching a $(p,q)$-cable space. We apply our results to a conjecture of Chen and Yang which relates the asymptotics of the Turaev–Viro invariants to the simplicial volume of a compact oriented $3$-manifold. For $p$ and $q$ coprime, we show that the Chen–Yang volume conjecture is stable under $(p,q)$-cabling. We achieve our results by studying the linear operator ${\mathrm{\text{RT}}}_r$ associated to the torus knot cable spaces by the Reshetikhin–Turaev ${\mathrm{\text{SO}}}_3$-Topological Quantum Field Theory (TQFT), where the TQFT is well-known to be closely related to the desired Turaev–Viro invariants. In particular, our utilized method relies on the invertibility of the linear operator for which we provide necessary and sufficient conditions.

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.

Given a Heegaard splitting of a closed 3-manifold, the skein modules of the two handlebodies are modules over the skein algebra of their common boundary surface. The zeroth Hochschild homology of the skein algebra of a surface with coefficients in the tensor product of the skein modules of two handlebodies is interpreted as the skein module of the 3-manifold obtained by gluing the two handlebodies together along this surface. A spectral sequence associated to the Hochschild complex is constructed and conditions are given for the existence of algebraic torsion in the completion of the skein module of this 3-manifold.

We introduce the virtual magnetic Kauffman bracket skein module, and show how to find a skein relation for the f-polynomial through the module. Furthermore, we give a unified approach to constructions of skein relations which hold under certain conditions.

It is a well known result that the Jones polynomial of a non-split alternating link is alternating. We find the right generalization of this result to the case of non-split alternating tangles. More specifically: the Jones polynomial of tangles is valued in a certain skein module; we describe an alternating condition on elements of this skein module, show that it is satisfied by the Jones invariant of the single crossing tangles (⤲) and (⤲), and prove that it is preserved by appropriately "alternating" planar algebra compositions. Hence, this condition is satisfied by the Jones polynomial of all alternating tangles. Finally, in the case of 0-tangles, that is links, our condition is equivalent to simple alternation of the coefficients of the Jones polynomial.

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.

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).

We classify non-affine, prime knots in the solid torus up to 6 crossings. We establish which of these are amphicheiral: almost all knots with symmetric Jones polynomial are amphicheiral, but in a few cases we use stronger invariants, such as HOMFLYPT and Kauffman skein modules, to show that some such knots are not amphicheiral. Examples of knots with the same Jones polynomial that are different in the HOMFLYPT skein module are presented. It follows from our computations, that the wrapping conjecture is true for all knots up to 6 crossings.

For a ring R, we denote by the free R-module spanned by the isotopy classes of singular links in 𝕊³. Given two invertible elements x, t ∈ R, the HOMFLY-PT skein module of singular links in 𝕊³ (relative to the triple (R, t, x)) is the quotient of by local relations, called skein relations, that involve t and x. We compute the HOMFLY-PT skein module of singular links for any R such that (t^-1 - t + x) and (t^-1 - t - x) are invertible. In particular, we deduce the Conway skein module of singular links.

In [On the HOMFLY-PT skein module of S¹ × S², Math. Z.237(4) (2001) 769–814], Gilmer and Zhong established the existence of an invariant for links in S¹ × S² which is a rational function in variables a and s and satisfies the HOMFLY-PT skein relations. We give formulas for evaluating this invariant in terms of a standard, geometrically simple basis for the HOMFLY-PT skein module of the solid torus. This allows computation of the invariant for arbitrary links in S¹ × S² and shows that the invariant is in fact a Laurent polynomial in a and z = s – s^-1. Our proof uses connections between HOMFLY-PT skein modules and invariants of Legendrian links. As a corollary, we extend HOMFLY-PT polynomial estimates for the Thurston–Bennequin number to Legendrian links in S¹ × S² with its tight contact structure.

Using Chebyshev polynomials, C. Frohman and R. Gelca introduced a basis of the Kauffman bracket skein module of the torus. This basis is especially useful because the Jones–Kauffman product can be described via a very simple Product-to-Sum formula. Presented in this work is a diagrammatic proof of this formula, which emphasizes and demystifies the role played by Chebyshev polynomials.

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.

In this paper, the properties of the Kauffman bracket skein module (KBSM) of $L(p,q)$ are investigated. Links in lens spaces are represented both through band and disk diagrams. The possibility to transform between the diagrams enables us to compute the KBSM on an interesting class of examples consisting of inequivalent links with equivalent lifts in the $3$-sphere. The computations show that the KBSM is an essential invariant, that is, it may take different values on links with equivalent lifts. We also show how the invariant is related to the Kauffman bracket of the lift in the $3$-sphere.

Based on the presentation of the Kauffman bracket skein algebra of the thickened torus given by the third author in previous work [4], Frohman and Gelca established a complete description of the multiplicative operation leading to a famous product-to-sum formula. In this paper, we study the multiplicative structure of the Kauffman bracket skein algebra of the thickened four-holed sphere. We present an algorithm to compute the product of any two elements of the algebra, and give an explicit formula for some families of curves. We surmise that the algorithm has quasi-polynomial growth with respect to the number of crossings of a pair of curves. Further, we conjecture the existence of a positive basis for the algebra.

Conference like this in Dallas is a perfect occasion to give some reminiscences of my 40 years of work as a mathematician, and to muse a little about my mathematical past. It is a rare occasion to be a little sentimental. In this essay, I summarize and expand my birthday talk given at University of Texas at Dallas conference. There, I mostly covered the early years of my mathematical career ending with a detailed description about the smallest volume hyperbolic 3-manifold and the HOMFLYPT polynomial. Thus I seldom venture to the time after 1984 (for example to mention skein modules constructed in April 1987) and only occasionally after 1995 when I settled to work at George Washington University and have had several successful PhD students.

We extend the Jones polynomial for links in S³ to links in L(p, q), p>0. Specifically, we show that the (2, ∞)-skein module of L(p, q) is free with [p/2]+1 generators. In the case of S¹×S² the skein module is infinitely generated.

To any orientable 3-manifold one can associate a module, called the (2, ∞)-skein module, which is essentially a generalization of the Jones polynomial of links in S³. For an uncountable collection of open contractible 3-manifolds, each constructed in a fashion similar to the classic Whitehead manifold, we prove that their (2, ∞)-skein modules are infinitely generated, torsion free, but not free. These examples stand in stark contrast to , whose (2, ∞)-skein module is free on one generator. To each of these manifolds we associate a subgroup G of the rationals which may be interpreted via wrapping numbers. We show that the skein module of M has a natural filtration by modules indexed by G. For the specific case of the Whitehead manifold, we describe its (2, ∞)-skein module and associated filtration in greater detail.