Related papers: Extending Quasi-Alternating Links
Since the 1974 paper by Peskine and Szpiro, liaison theory via complete intersections, and more generally via Gorenstein varieties, has become a standard tool kit in commutative algebra and algebraic geometry, allowing to compare algebraic…
A virtual link is said to be almost classical (AC) if it has a homologically trivial representative in some thickened surface $\Sigma \times [0,1]$, where $\Sigma$ is a closed orientable surface. AC links provide a useful window for…
Among non-alternating knots with $n\le 12$ crossings given in \cite{1} Turaev genus is not known for 191 knot. For 154 of them we show that they are almost alternating, so their Turaev genus is 1.
Generalizing unknotting number, $n$-adjacent knots have $n$ crossings such that changing any non-empty subset of them results in the unknot. In this paper, we determine the 2-adjacent knots through 12 crossings. Using Heegaard Floer…
Link equivalence up to isotopy in a 3-space is the problem that lies at the root of knot theory, and is important in 3-dimensional topology and geometry. We consider its restriction to alternating links, given by two alternating diagrams…
Using the colored Kauffman skein relation, we study the highest and lowest $4n$ coefficients of the $n^{th}$ unreduced colored Jones polynomial of alternating links. This gives a natural extension of a result by Kauffman in regard with the…
In this paper I give estimates for the minimal crossing number, leading to a short proof that the crossing number is additive for torus links. These estimates are applied to several classes of links. Finally, I prove a part of a conjecture…
We prove that the detection rate of n-crossing alternating links by many standard link invariants decays exponentially in n, implying that they detect alternating links with probability zero. This phenomenon applies broadly, in particular…
We proved by computer enumeration that the Jones polynomial distinguishes the unknot for knots up to 22 crossings. Following an approach of Yamada, we generated knot diagrams by inserting algebraic tangles into Conway polyhedra, computed…
Knot Theory is currently a very broad field. Even a long survey can only cover a narrow area. Here we concentrate on the path from Goeritz matrices to quasi-alternating links. On the way, we often stray from the main road and tell related…
The Jones polynomial of an alternating link is a certain specialization of the Tutte polynomial of the (planar) checkerboard graph associated to an alternating projection of the link. The Bollobas-Riordan-Tutte polynomial generalizes the…
We construct geometrically two universal link invariants: universal ADO invariant and universal Jones invariant, as limits of invariants given by graded intersections in configuration spaces. More specifically, for a fixed level $\mathscr…
The theory of the Kauffman bracket, which describes the Jones polynomial as a sum over closed circles formed by the planar resolution of vertices in a knot diagram, can be straightforwardly lifted from sl(2) to sl(N) at arbitrary N -- but…
We prove that the Khovanov homology of alternating knots and 2-component links is equal (as a singly graded group) to the singular homology of a certain space of trace- free, binary dihedral representations of the link group. More…
We prove that any link admitting a diagram with a single negative crossing is strongly quasipositive. This answers a question of Stoimenow's in the (strong) positive. As a second main result, we give simple and complete characterizations of…
We introduce a framework to analyze knots and links in an unmarked solid torus. We discuss invariants that detect when such links are equivalent under an ambient homeomorphism, and show that the multivariable Alexander polynomial is such in…
As an extension of positive and almost positive diagrams and links, we study two classes of links we call successively almost positive and weakly successively almost positive links. We prove various properties of polynomial invariants and…
We prove a simple necessary and sufficient condition for a two-bridge knot K(p,q) to be quasipositive, based on the continued fraction expansion of p/q. As an application, coupled with some classification results in contact and symplectic…
This paper introduces a new algebra, the crossing algebra, that is applied to count the number of components for arborescent knots, links, tangles or states (of a state polynomial expansion such as the Kauffman bracket). This algebra is…
The Tait conjecture states that alternating reduced diagrams of links in S^3 have the minimal number of crossings. It has been proved in 1987 by M. Thistlethwaite, L. Kauffman and K. Murasugi studying the Jones polynomial. The author proved…