arXiv research

Correspondence found between Askey-Wilson polynomials and genus-two handlebody skein module.

Generalizes positivity conjecture to Roger--Yang skein algebras using polynomials.

New skein theory for Links-Gould polynomial simplifies link evaluations.

The paper improves bounds on the complexity of computing link polynomials.

Paper introduces a new skein relation for multivariable polynomials of virtual links.

Polynomial algorithm for multiplication on one-hole torus skein algebra.

The paper improves bounds on skein tree depth and delta-crossing numbers for knots and links.

Extends A-type coefficient polynomials to B-type setting, introducing new invariants.

Worldsheet skein D-module for Hopf link conormal uniquely determines partition functions.

Study $G_2$ skein algebra elements using Kuperberg webs and threading operations.

In this paper, we study the properties of the colored HOMFLY polynomials via HOMFLY skein theory. We prove some limit behaviors and symmetries of the colored HOMFLY polynomial predicted in some physicists' recent works.

Quantum theory constructs a group and skein module for knot complements.

We introduce new skein invariants of links based on a procedure where we first apply the skein relation only to crossings of distinct components, so as to produce collections of unlinked knots. We then evaluate the resulting knots using a given invariant. A skein invariant can be computed on each link solely by the use…

Following the recent work by Chan, and by Morton and Hadji on the Homflypt polynomials of some generalized Hopf links, we investigate the Kauffman polynomials of generalized Hopf links. By studying the Kauffman skein module of the solid torus S^1\times D^2, we establish a similar skein map on the Kauffman skein module …

In this note we define a polynomial invariant for colored links by a skein relation. It specializes to the Jones polynomial for classical links.

We give characterizations of the skein polynomial for links (as well as Jones and Alexander-Conway polynomials derivable from it), avoiding the usual "smoothing of a crossing" move. As by-products we have characterizations of these polynomials for knots, and for links with any given number of components.

We introduce higher skein modules of links generalizing the Conway skein module. We show that these modules are closely connected to the HOMFLY polynomial.

A new polynomial invariant for strongly involutive links.

Study Type $C$ skein modules using $Sp(2n)$ webs and construct transparent elements.

Paper computes skein modules of 3-manifolds using braids.

Extends Kauffman's formula to 3-manifolds with markings.

New invariants of links are constructed using the skein invariant polynomial of colored links defined by the author in [1]. These invariants are stronger than the homflypt polynomial.

Paper compares skein modules to Kauffman bracket modules.

Given any oriented link diagram, one can construct knot invariants using skein relations. Usually such a skein relation contains three or four terms. In this paper, the author introduces several new ways to smooth a crossings, and uses a system of skein equations to construct link invariant. This invariant can also be …

The Jones polynomial is a famous link invariant that can be defined diagrammatically via a skein relation. Khovanov homology is a richer link invariant that categorifies the Jones polynomial. Using spectral sequences, we obtain a skein-type relation satisfied by the Khovanov homology. Thanks to this relation, we are ab…

Kuperberg introduced web spaces for some Lie algebras which are generalizations of the Kauffman bracket skein module on a disk with marked points. We derive some formulas for $A_1$ and $A_2$ clasped web spaces by graphical calculus using skein theory. These formulas are colored version of skein relations, twist formula…

Colored HOMFLY-PT invariant, the generalization of the colored Jones polynomial, is one of the most important quantum invariants of links. This paper is devoted to investigating the basic structures of the colored HOMFLY-PT invariants of links. By using the HOMFLY-PT skein theory, firstly, we show that the (reformulate…

We prove the existence of a polynomial invariant that satisfies the HOMFLY skein relation for links in a lens space. In the process we also develop a skein theory of toroidal grid diagrams in a lens space.

The paper classifies virtual knot polynomials and trivalent graph invariants using skein theory.

The paper shows the computation of the noncommutative generalization of the A-polynomial of the trefoil knot. The classical A-polynomial was introduced by Cooper, Culler, Gillet, Long and Shalen, and was generalized to the context of Kauffman bracket skein modules by the author in joint work with Frohman and Lofaro. A …

The aim of this paper is to define two link invariants satisfying cubic skein relations. In the hierarchy of polynomial invariants determined by explicit skein relations they are the next level of complexity after Jones, HOMFLY, Kauffman and Kuperberg's $G_2$ quantum invariants. Our method consists in the study of Mark…

Quantum Frobenius map for $SL_3$ skein modules constructed and described.

Quantum approach to volume computation from colored Jones polynomials.

Researchers solved a conjecture about a mathematical structure of connected sums of solid tori.

Study of cubic skein modules in 3-sphere and arbitrary 3-manifolds.

New proof of SL(n) skein algebra for twice punctured sphere, showing it's a polynomial algebra.

Researchers compute the Kauffman bracket skein module of a specific 3-manifold.

A new algebraic method for computing helicity is developed, by discovering a relationship between helicity of fluid mechanics and algebraic polynomial invariants of knot theory. We have constructed a topological invariant $t^{H\left(\mathcal{L}\right)}$ for a link $\mathcal{L}$ of knots, where $H$ is the helicity of a …

New knot polynomials yield simple results modulo primes.

Modified proof for a broader class of links.

We show that the if a sequence of normalized polynomials gives rise to a positive basis of the skein algebra of a surface, then it is sandwiched between the two types of Chebyshev polynomials. For the closed torus, we show that the normalized sequence of Chebyshev polynomials of type one $(\hat{T}_n)$ is the only one w…

Counterexample disproves conjecture about 3-manifold modules.

Kauffman knot polynomial invariants are discovered in classical abelian Chern-Simons field theory. A topological invariant $t^{I\left( \mathcal{L} \right) }$ is constructed for a link $\mathcal{L}$, where $I$ is the abelian Chern-Simons action and $t$ a formal constant. For oriented knotted vortex lines, $t^{I}$ satisf…

We give a simple and practical algorithm to compute the link polynomials, which are defined according to the skein relations. Our method is based on a new total order on the set of all braid representatives. As by-product a new complete link invariant are obtained.

We show that the Chebyshev polynomials form a basic block of any positive basis of the Kauffman bracket skein algebras of surfaces.

We introduce the deformed fermionic numbers, corresponding to the skein relations, the main characteristics of knots and links. These fermionic numbers allow one to restore the skein relations. For the Alexander (Jones) skein relation we introduce corresponding Alexander (Jones) fermionic q-numbers, and for the HOMFLY …

Develops skein theory for 3-manifolds with defects, extending quantum character stacks.

Skein theory aids in computing YB homology and cohomology groups.