arXiv research

The abstract introduces a sequence of moves between two types of Heegaard diagrams for knot Floer homology.

This paper contains general formulae for the reduced relative Tutte, Kauffman bracket and Jones polynomials of families of virtual knots and links given in Conway notation and discussion of a counterexample to the Z-move conjecture of Fenn, Kauffman and Manturov.

Paper defines new invariants for surface-links using graph diagrams and magmas.

We classify the Montesinos links up to mutation and 5-move equivalence, and obtain from this a Jones and Kauffman polynomial test for a Montesinos link.

We start a systematic analysis of links up to 5-move equivalence. Our motivation is to develop tools which later can be used to study skein modules based on the skein relation being deformation of a 5-move (in an analogous way as the Kauffman skein module is a deformation of a 2-move, i.e. a crossing change). Our main …

The paper computes the Kauffman bracket skein module of $S^1 imes S^2$ via braids.

Classifies virtual links up to a specific move.

New method for computing Kauffman bracket skein module of lens spaces using unoriented braids.

In this paper, it is shown that there are no nonconstant Goussarov-Polyak-Viro finite-type invariants that are invariant under the virtualization move. As an immediate corollary, we obtain the theorem which states none of the Birman coefficients of the Jones-Kauffman polynomial are of GPV finite type.

Diagrams and Reidemeister moves for links in a twisted S^1-bundle over an unorientable surface are introduced. Using these diagrams, we compute the Kauffman Bracket Skein Module (KBSM) of the connected sum of two projective spaces. In particular, we show that it has torsion. We also present a new computation of the KBS…

A pass-move and a $#$-move are local moves on oriented links defined by L.H. Kauffman and H. Murakami respectively. Two links are self pass-equivalent (resp. self $#$-equivalent) if one can be deformed into the other by pass-moves (resp. $#$-moves), where non of them can occur between distinct components of the link. T…

The paper extends knot polynomials to annular and toroidal pseudo links.

The paper computes the Kauffman bracket skein module of $(2, 2p+1)$-torus knots using braids.

If a rectangular diagram represents the trivial knot, then it can be deformed into the trivial rectangular diagram with only four edges by a finite sequence of merge operations and exchange operations, without increasing the number of edges, which was shown by I. A. Dynnikov. Using this, Henrich and Kauffman gave an up…

L. Kauffman conjectured that a particular solution of the Chinese Rings puzzle is the simplest possible. We prove his conjecture by using low-dimensional topology and group theory. We notice also a surprising connection between the Chinese Rings and Habiro moves (related to Vassiliev invariants).

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

Survey on link diagrams in Seifert manifolds and skein modules.

New bases for Kauffman bracket skein module of genus 2 handlebody.

Method converts virtual link diagrams to normal ones.

Alexander polynomial linked to twist sites in rational links.

We present an invariant of connected and oriented closed 3-manifolds based on a coribbon Weak Hopf Algebra H with a suitable left-integral. Our invariant can be understood as the generalization to Weak Hopf Algebras of the Hennings-Kauffman-Radford evaluation of an unoriented framed link using a dual quantum-trace. Thi…

In order to obtain a Markov theorem without stabilization, Birman and Menasco introduced the notion of exchange related braids. In this paper I study the way the Fiedler polynomial distinguishes conjugacy classes of some particular braided knots. I introduce the Kauffman bracket in the solid torus. Its Taylor expansion…

It is a natural question to ask whether two links are equivalent by the following moves -- parallel parts of a link are changed to k-times half-twisted parts and if they are, how many moves are needed to go from one link to the other. In particular if k=2 and the second link is a trivial link it is the question about t…

We introduce diagrams and Reidemeister moves for links in FxS^{1}, where F is an orientable surface. Using these diagrams we compute (in a new way) the Kauffman Bracket Skein Modules (KBSM) for D^{2}xS^{1} and AxS^{1}, where D^{2} is a disk and A is an annulus. Moreover, we also find the KBSM for the F_{0,3}xS^{1}, whe…

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 Reidemeis…

Khovanov defined graded homology groups for links L in R^3 and showed that their polynomial Euler characteristic is the Jones polynomial of L. Khovanov's construction does not extend in a straightforward way to links in I-bundles M over surfaces F not D^2 (except for the homology with Z/2 coefficients only). Hence, the…

The paper characterizes discrete Morse functions on knot diagrams and generalizes a clock theorem.

Rectangular mosaics extend virtual knot studies to larger polygons.

ReAPR simplifies hard unknots by reembedding and rerouting, revealing hidden simplifications.

Positive basis of Kauffman bracket skein algebras proven using Chebyshev polynomials.

Extend Kauffman bracket skein module to homology theory using Heegaard splittings

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 …

Formulae for Vassiliev invariants derived from Kauffman polynomial.

We extend a mod 2 relation between the Kauffman and Homfly polynomials, first observed by Rudolph in 1987, to the general Kauffman and Homfly satellite invariants.

Samuel J. Lomonaco Jr and Louis H. Kauffman conjectured that tame knot theory and knot mosaic theory are equivalent. We give a proof of the Lomonaco-Kauffman conjecture.

Paper disproves a theorem about Kauffman bracket skein module structure.

In this paper, we investigate twist sequences for Kauffman finite-type invariants and Goussarov-Polyak-Viro finite-type invariants. It is shown that one obtains a Kauffman or GPV type of degree $\le n$ if and only if an invariant is a polynomial of degree $\le n$ on every twist lattice of the right form. The main resul…

New invariants for tied links extend classical polynomial invariants.

Paper compares skein modules to Kauffman bracket modules.

Study Kauffman module at -1+ε, conjectures relation to Reidemeister torsion.

The paper explores knotoids, pseudo knotoids, braidoids, and pseudo braidoids on the torus.

The paper constructs new surface-link invariants using link invariants and applies them to the Kauffman bracket.

The Kauffman bracket polynomial is calculated for specific Turk's head knots.

We prove the following: Let $2p + 1$ be no less than 5 and $p$ be a natural number. Let $K$ and $J$ be closed, oriented, $(2p+1)$-dimensional connected, $(p-1)$-connected, simple submanifolds of the standard $(2p+3)$-sphere. Then $K$ is equivalent to $J$ if and only if a Seifert matrix associated with a simple Seifert …

Alternative basis for Kauffman bracket skein module of solid torus found using braids.

Generalizes Kauffman-Vogel polynomials to oriented and unoriented 4-valent graphs.

Formula for Dehn twists in skein algebras defined on surfaces.

The W-polynomial is applied in two ways to questions involving the Kauffman bracket of some families of links. First we find a geometric property of a link diagram, which is less than or equal to the twist number, that bounds the Mahler measure of the Kauffman bracket. Second we find a general form for the Kauffman bra…