Related papers: Augmentations, annuli, and Alexander polynomials
We define an SFT-type invariant for Legendrian knots in the standard contact $\mathbb{R}^3$. The invariant is a deformation of the Chekanov-Eliashberg differential graded algebra. The differential consists of a part that counts index zero…
Given a homomorphism from a link group to a group, we introduce a $K_1$-class in another way, which is a generalization of the 1-variable Alexander polynomial. We compare the $K_1$-class with $K_1$-classes in \cite{Nos} and with…
The derived group of a permutation representation, introduced by R.H. Crowell, unites many notions of knot theory. We survey Crowell's construction, and offer new applications. The twisted Alexander group of a knot is defined. Using it, we…
Two knots in three-space are S-equivalent if they are indistinguishable by Seifert matrices. We show that S-equivalence is generated by the doubled-delta move on knot diagrams. It follows as a corollary that a knot has trivial Alexander…
The perturbed Alexander invariant $\rho_1$, defined by Bar-Natan and van der Veen, is a powerful, easily computable polynomial knot invariant with deep connections to the Alexander and colored Jones polynomials. We study the behavior of…
We give necessary and sufficient conditions for an integer to be the signature of a (4q-1)-knot in the (4q+1)-sphere with a given square-free Alexander polynomial.
Given a transverse knot $K$ in a three dimensional contact manifold $(Y,\alpha)$, in [13] Colin, Ghiggini, Honda and Hutchings define a hat version of embedded contact homology for $K$, that we call $\widehat{ECK}(K,Y,\alpha)$, and…
The Alexander biquandle of a virtual knot or link is a module over a 2-variable Laurent polynomial ring which is an invariant of virtual knots and links. The elementary ideals of this module are then invariants of virtual isotopy which…
This paper deals with the study of a new family of knot invariants: the $L^2$-Alexander invariant. A main result is to give a method of computation of the $L^2$-Alexander invariant of a knot complement using any presentation of default 1 of…
We define a "reduced" version of the knot Floer complex $CFK^-(K)$, and show that it behaves well under connected sums and retains enough information to compute Heegaard Floer $d$-invariants of manifolds arising as surgeries on the knot…
It is well-known:Suppose there are three 1-dimensional links $K_+$, $K_-$, $K_0$ such that $K_+$, $K_-$, and $K_0$ coincide out of a 3-ball $B$ trivially embedded in $S^3$ and that $K_+\cap B$, $K_-\cap B$, and $K_0\cap B$ are drawn as…
In these notes we collect some results about finite dimensional representations of $U_q(\mathfrak{gl}(1|1))$ and related invariants of framed tangles which are well-known to experts but difficult to find in the literature. In particular, we…
Let $K$ be a genus $g$ alternating knot with Alexander polynomial $\Delta_K(T)=\sum_{i=-g}^ga_iT^i$. We show that if $|a_g|=|a_{g-1}|$, then $K$ is the torus knot $T_{2g+1,\pm2}$. This is a special case of the Fox Trapezoidal Conjecture.…
Using an invariant defined by Rasmussen, we extend an argument given by Hedden and Watson which further restricts which Alexander polynomials can be realized by $L$-space knots.
We discuss a matrix of periodic holomorphic functions in the upper and lower half-plane which can be obtained from a factorization of an Andersen-Kashaev state integral of a knot complement with remarkable analytic and asymptotic properties…
A perturbative expansion of knot invariants is derived using quantum cluster algebras. By interpreting the $R$-matrix of $U_q(\mathfrak{sl}_2)$ as a cluster transformation and introducing an auxiliary parameter $\epsilon$, we derive a…
In this note we use Heegaard Floer homology to study smooth cobordisms of algebraic knots and complex deformations of cusp singularities of curves. The main tool will be the concordance invariant $\nu^+$: we study its behaviour with respect…
The $n$-loop Kontsevich invariant of knots takes its value in the completion of the space of $n$-loop open Jacobi diagrams, which is an infinite dimensional vector space. Since the 1-loop part is presented by the Alexander polynomial, we…
We study the twisted Alexander polynomial $\Delta_{K,\rho}$ of a knot $K$ associated to a non-abelian representation $\rho$ of the knot group into $SL_2(\BC)$. It is known for every knot $K$ that if $K$ is fibered, then for every…
This article provides an overview of relative strengths of polynomial invariants of knots and links, such as the Alexander, Jones, Homflypt, Kaufman two-variable polynomial, and Khovanov polynomial.