Related papers: Knot polynomials in the first non-symmetric repres…

Given a knot K in the 3-sphere, consider a singular disk bounded by K and the intersections of K with the interior of the disk. The absolute number of intersections, minimised over all choices of singular disk with a given algebraic number…

We show that, for each integer n, there exist infinitely many pairs of n-framed knots representing homeomorphic but non-diffeomorphic (Stein) 4-manifolds, which are the simplest possible exotic 4-manifolds regarding handlebody structures.…

We describe a new method of producing equations for the canonical component of representation variety of a knot group into $PSL_2(\mathbb{C})$. Unlike known methods, this one does not involve any polyhedral decomposition or triangulation of…

Families of alternating knots (links) and tangles are studied using as building block the conway defined as the twisting of two strands. The regular representation of knots assumes the projection has the minimal number of overpassings, and…

For a graph G embedded in an orientable surface \Sigma, we consider associated links L(G) in the thickened surface \Sigma \times I. We relate the HOMFLY polynomial of L(G) to the recently defined Bollobas-Riordan polynomial of a ribbon…

Suppose the knot group G(K) of a knot K has a non-abelian representation \rho on A_4 \subset GL(4,Z). We conjecture that the twisted Alexander polynomial of K associated to \rho is of the form: \Delta_K(t)/(1-t) \phi(t^3), where \Delta_K…

We calculate limits of the colored Jones polynomials of the figure-eight knot and conclude that in most cases they determine the volumes and the Chern--Simons invariants of the three-manifolds obtained by Dehn surgeries along it.

Whitehead doubles provide a plethora of examples of knots that are topologically slice but not smoothly slice. We discuss the problem of the Whitehead double of the Figure 8 knot and survey commonly used techniques to obstructing sliceness.…

We prove for the first time that knot Floer homology and Khovanov homology can detect non-fibered knots, and that HOMFLY homology detects infinitely many such knots; these theories were previously known to detect a mere six knots, all…

This survey explores knot polynomials and their categorification, culminating in the homological invariants of knots. We begin with an overview of classical knot polynomials, progressing towards the superpolynomial and its role in unifying…

A Chebyshev curve C(a,b,c,\phi) has a parametrization of the form x(t)=Ta(t); y(t)=T_b(t) ; z(t)= Tc(t + \phi), where a,b,c are integers, Tn(t) is the Chebyshev polynomial of degree n and \phi \in \RR. When C(a,b,c,\phi) has no double…

This work presents formulas for the Kauffman bracket and Jones polynomials of 3-bridge knots using the structure of Chebyshev knots and their billiard table diagrams. In particular, these give far fewer terms than in the Skein relation…

We introduce a new algebraic topological technique to detect non-fibred knots in the three sphere using the twisted Alexander invariants. As an application, we show that for any Seifert matrix of a knot with a nontrivial Alexander…

We prove that for knots, the evaluation of the Jones polynomial at the sixth root of unity, as well as the evaluation of the $Q$-polynomial at the reciprocal of the golden ratio, are uniquely determined by the oriented homeomorphism type of…

In 1979, Hartley and Kawauchi proved that the Conway polynomial of a strongly negative amphichiral knot factors as $f(z)f(-z)$. In this paper, we normalize the factor $f(z)$ to define the half-Conway polynomial. First, we prove that the…

By a recent result of Livingston, it is known that if a knot has a prime power branched cyclic cover that is not a homology sphere, then there is an infinite family of non-concordant knots having the same Seifert form as the knot. In this…

We define a family of quiver representation-valued invariants of oriented classical and virtual knots and links associated to a choice of finite quandle $X$, abelian group $A$, set of quandle 2-cocycles $C\subset H^2_Q(x;A)$, choice of…

A knot K in the 3-sphere is superslice if there is a slice disk D in the 4-ball such that the double of D along K is the unknotted 2-sphere S in $S^4$. Answering a question of Livingston-Meier, we find smoothly slice (in fact doubly slice)…

Twisted torus knots are a generalization of torus knots, obtained by introducing additional full twists to adjacent strands of the torus knots. In this article, we present an explicit formula for the Alexander polynomial of twisted torus…

Neumann and Reid conjecture that there are exactly three knot complements which admit hidden symmetries. This paper establishes several results that provide evidence for the conjecture. Our main technical tools provide obstructions to…