The paper introduces the Slope Conjecture which relates the degree of the Jones polynomial of a knot and its parallels with the slopes of incompressible surfaces in the knot complement. More precisely, we introduce two knot invariants, the Jones slopes (a finite set of rational numbers) and the Jones period (a natural number) of a knot in 3-space. We formulate a number of conjectures for these ...

The Volume Conjecture loosely states that the limit of the n-th colored Jones polynomial of a hyperbolic knot, evaluated at the primitive complex n-th root of unity is a sequence of complex numbers that grows exponentially. Moreover, the exponential growth rate is proportional to the hyperbolic volume of the knot. We provide an efficient formula for the colored Jones function of the simplest hy...

For a hyperbolic knot, the excellent curve is the component of the character variety containing the complete hyperbolic structure on the complement of the knot. In this paper we introduce the concept of net presentation of a knot with 2n strings which generalises the well known plat presentation of a knot with 2n strings. Nets with 4 strings are the correct setting to apply quaternion methods t...

We establish a fundamental connection between smooth and polygonal knot energies, showing that the Minimum Distance Energy for polygons inscribed in a smooth knot converges to the Möbius Energy of the smooth knot as the polygons converge to the smooth knot. However, the polygons must converge in a “nice” way, and the energies must be correctly regularized. We determine an explicit error bound b...

We explore the p-colorability of a family of knots known as weaving knots. First, we show some general results pertaining to p-coloring any (m, n) weaving knot. We then determine the p-colorability of any (m, 3) weaving knot, and designate these p-colorings into two separate types, whose characteristics we begin to explore. We also partially classify some of the p-colorings for any (m, n) weavi...

Suppose M is a closed irreducible orientable 3-manifold, K is a knot in M , P and Q are bridge surfaces for K and K is not removable with respect to Q. We show that either Q is equivalent to P or d(K,P ) ≤ 2 − χ(Q − K). If K is not a 2-bridge knot, then the result holds even if K is removable with respect to Q. As a corollary we show that if a knot in S has high distance with respect to some br...

This paper describes how a subclass of the rational knots* may be constructed sequentially., the knots in the sequence having 19 29 ..., i s ... crossings. For these knots, the values of a certain knot invariant are Fibonacci numbers, the i knot in the sequence having invariant number Fi . The knot invariant has a wide number of interpretations and properties, and some of these will be outlined...

The action of various DNA topoisomerases frequently results in characteristic changes in DNA topology. Important information for understanding mechanistic details of action of these topoisomerases can be provided by investigating the knot types resulting from topoisomerase action on circular DNA forming a particular knot type. Depending on the topological bias of a given topoisomerase reaction,...

Manolescu, Ozsváth and Sarkar gave [9] an explicit description of knot Floer homology for a knot in the three-sphere as the homology groups of a chain complex CK which is described in terms of the combinatorics of a grid diagram for a knot. In fact, the constructions of [9] are done with coefficients in Z=2Z; a lift of these constructions to coefficients in Z is given by Manolescu and the autho...

We initiate the study of classical knots through the homotopy class of the nth evaluation map of the knot, which is the induced map on the compactified n-point configuration space. Sending a knot to its nth evaluation map realizes the space of knots as a subspace of what we call the nth mapping space model for knots. We compute the homotopy types of the first three mapping space models, showing...