Related papers: Negative amphichiral knots and the half-Conway pol…
Homologically fibered knots are knots whose exteriors satisfy the same homological conditions as fibered knots. In our previous paper, we observed that for such a knot, higher-order Alexander invariants defined by Cochran, Harvey and Friedl…
Let $K$ be a knot type for which the quadratic term of the Conway polynomial is nontrivial, and let $\gamma: \mathbb{R}\to \mathbb{R}^3$ be an analytic $\mathbb{Z}$-periodic function with non-vanishing derivative which parameterizes a knot…
By considering a particular type of invariant Seifert surfaces we define a homomorphism $\Phi$ from the (topological) equivariant concordance group of directed strongly invertible knots $\widetilde{\mathcal{C}}$ to a new equivariant…
We use matchings on Lyndon words to classify flat knots up to 8 crossings. Using flat knots invariants such as the based matrix, the $\phi$-invariant, the flat arrow polynomial, and the flat Jones-Krushkal polynomial, we distinguish all…
We use the 2-loop term of the Kontsevich integral to show that there are (many) knots with trivial Alexander polynomial which don't have a Seifert surface whose genus equals the rank of the Seifert form. This is one of the first…
The motivation for this work was to construct a nontrivial knot with trivial Jones polynomial. Although that open problem has not yielded, the methods are useful for other problems in the theory of knot polynomials. The subject of the…
Associated with each oriented link is the two variable Homflypt polynomial. The Morton-Franks-Williams (MFW) inequality gives rise to an expression for the Homflypt polynomial with MFW coefficient polynomials. These MFW coefficient…
Murasugi discovered two criteria that must be satisfied by the Alexander polynomial of a periodic knot. We generalize these to the case of twisted Alexander polynomials. Examples demonstrate the application of these new criteria, including…
In this paper we discuss an approach to calculate knot polynomials on a photonic processor. Calculations of knot polynomials is a computationally difficult problem and therefore it is interesting to use new advanced calculation methods to…
We study pseudo-classical knots in the non-orientable thickening of a non-orientable surface, specifically knots that are orientation-preserving paths in a non-orientable $3$-manifold of the form (non-orientable surface) $\times$ $[0, 1]$.…
We prove that certain fibered, $-$amphicheiral knots are rationally slice. Moreover, we show that the concordance invariants $\nu^+$ and $\Upsilon(t)$ from Heegaard Floer homology vanish for a class of knots that includes rationally slice…
Besides offering a friendly introduction to knot homologies and quantum curves, the goal of these lectures is to review some of the concrete predictions that follow from the physical interpretation of knot homologies. In particular, this…
We consider diagrams of links in $S^2$ obtained by projection from $S^3$ with the Hopf map and the minimal crossing number for such diagrams. Knots admitting diagrams with at most one crossing are classified. Some properties of these knots…
We prove that 0 is a characterizing slope for infinitely many knots, namely the genus-1 knots whose knot Floer homology is 2-dimensional in the top Alexander grading, which we classified in recent work and which include all $(-3,3,2n+1)$…
The colored HOMFLY polynomials, which describe Wilson loop averages in Chern-Simons theory, possess an especially simple representation for torus knots, which begins from quantum R-matrix and ends up with a trivially-looking split W…
We show that for genus one knots the Alexander polynomial and the homology of the double cover branching over the knot provide obstructions to cosmetic crossings. As an application we prove the nugatory crossing conjecture for the…
It was asked by J.Birman, Williams, and L.Rudolph whether nontrivial Lorentz knots have always positive signature. Lorentz knots are examples of positive braids (in our convention they have all crossings negative so they are negative…
An oriented link L in a 3-sphere S in complex 2-space is a C-boundary if it bounds a piece of algebraic curve in the 4-ball bounded by S. Using Kronheimer and Mrowka's proof of the Thom Conjecture, we construct many oriented knots which are…
We show that there are only finitely many homogeneous links whose Conway polynomial has any given degree. Using this we give an example of an inhomogeneous, fibred knot. Secondly, we show how to compute the monodromy of a homogeneous link…
In an earlier paper the first author defined a non-commutative A-polynomial for knots in 3-space, using the colored Jones function. The idea is that the colored Jones function of a knot satisfies a non-trivial linear q-difference equation.…