arXiv research

The paper explores basic properties of knot skein invariants.

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…

Introduces admissible skein modules for non-semisimple categories.

Develops a new spectrum for annular links, recovering a transverse invariant at extreme gradings.

The theory of link-homotopy, introduced by Milnor, is an important part of the knot theory, with Milnor's mu-bar-invariants being the basic set of link-homotopy invariants. Skein relations for knot and link invariants played a crucial role in the recent developments of knot theory. However, while skein relations for Al…

Solves a recursion for Gromov-Witten invariants of the unknot.

Skein lasagna module calculates 4-manifold invariants using handle decompositions.

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 …

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

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…

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.

Defines a new 2+1-G-HQFT using graded skein modules.

We introduce a new class of quantum enhancements we call biquandle brackets, which are customized skein invariants for biquandle colored links.Quantum enhancements of biquandle counting invariants form a class of knot and link invariants that includes biquandle cocycle invariants and skein invariants such as the HOMFLY…

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

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…

In this paper, we investigate the properties of the full colored HOMFLYPT invariants in the full skein of the annulus $\mathcal{C}$. We show that the full colored HOMFLYPT invariant has a nice structure when $q\rightarrow 1$. The composite invariant is a combination of the full colored HOMFLYPT invariants. In order to …

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

Constructs maps on skein modules using non-semisimple quantum invariants.

Proves a new skein exact triangle for real monopole Floer homology.

New categorifications of biquandle brackets defined.

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

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

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…

It is natural to try to place the new polynomial invariants of links in algebraic topology (e.g. to try to interpret them using homology or homotopy groups). However, one can think that these new polynomial invariants are byproducts of a new more delicate algebraic invariant of 3-manifolds which measures the obstructio…

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

New invariants derived from link homology for 4-manifolds.

Study of skein invariants on tori for various groups and quantum parameters.

Categorifies a skein relation for links colored by one-column Young diagrams.

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

By using the HOMFLY skein theory. We prove a strong integrality theorem for the reduced colored HOMFLYPT invariants defined by a basis in the full HOMFLY skein of the annulus.

Homomorphisms on quandle cohomology groups that raise the dimensions by one are studied in relation to the cocycle state-sum invariants of knots and knotted surfaces. Skein relations are also studied.

Skein modules are the main objects of an algebraic topology based on knots (or position). In the same spirit as Leibniz we would call our approach "algebra situs." When looking at the panorama of skein modules we see, past the rolling hills of homologies and homotopies, distant mountains - the Kauffman bracket skein mo…

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.

A new polynomial invariant for strongly involutive links.

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 …

Link homology theories connect to 4-manifold invariants and TQFTs.

We use recoupling theory to study the Kauffman bracket skein module of the quaternionic manifold over Z[A,A^{-1}] localized by inverting all the cyclotomic polynomials. We prove that the skein module is spanned by five elements. Using the quantum invariants of these skein elements and the Z_2 homology of the manifold, …

We define invariants for a framed link equipped with a SL2 local system in its complement and additional combinatorial data based on the theory of representations of stated skein algebras at roots of unity of punctured bigons and the geometric interpretation of their centers. The gauge invariance of the link invariant …

Traditionally introduced in terms of advanced topological constructions, many link invariants may also be defined in much simpler terms given their values on a few initial links and a recursive formula on a skein triangle. Then the crucial question to ask is how many initial values are necessary to completely determine…

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

The theory of welded and extended welded knots is a generalization of classical knot theory. Welded (resp. extended welded) knot diagrams include virtual crossings (resp. virtual crossings and wen marks) and are equivalent under an extended set of Reidemeister-type moves. We present a new class of invariants for welded…

Extends knotoid theory to multi-linkoids, studying various invariants.

We extend some results of Bonahon, Bullock, Turaev and Wong concerning the skein algebras of closed surfaces to L^e's stated skein algebra associated to open surfaces. We prove that the stated skein algebra with deforming parameter +1 embeds canonically into the centers of the stated skein algebras whose deforming para…

Algebra Situs is a branch of mathematics which has its roots in Jones' construction of his polynomial invariant of links and Drinfeld's work on quantum groups. It encompasses the theory of quantum invariants of knots and 3-manifolds, algebraic topology based on knots, operads, planar algebras, q-deformations, quantum g…

New invariant for singular links via bt-algebra.

Paper constructs a HOMFLYPT-type invariant for pseudo links.

In \cite{GZ}, Gilmer and Zhong established the existence of an invariant for links in $S^1\times S^2$ 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 ske…

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