Khovanov introduced a cohomology theory for oriented classical links whose graded Euler characteristic is the Jones polynomial. Since Khovanov’s theory is functorial for link cobordisms between classical links, we obtain an invariant of a surface-knot, called the Khovanov-Jacobsson number, by considering the surface-knot as a link cobordism between empty links. In this paper, we define an invar...

Conformally invariant functionals on the space of knots are introduced via extrinsic confor-mal geometry of the knot and integral geometry on the space of spheres. Our functionals are expressed in terms of a complex-valued 2-form which can be considered as the cross-ratio of a pair of infinitesimal segments of the knot. We show that our functionals detect the unknot as the total curvature does,...

Consider the Chern-Simons topological quantum field theory with gauge group SU2 and level k. Given a knot in the 3-sphere, this theory associates to the knot exterior an element in a vector space. We call this vector the knot state and study its asymptotic properties when the level is large. The latter vector space being isomorphic to the geometric quantization of the SU2-character variety of t...

HEEGAARD FLOER INVARIANTS AND CABLING Jennifer Hom Paul Melvin, Advisor A natural question in knot theory is to ask how certain properties of a knot behave under satellite operations. We will focus on the satellite operation of cabling, and on Heegaard Floertheoretic properties. In particular, we will give a formula for the Ozsváth-Szabó concordance invariant τ of iterated cables of a knot K in...

We study knotted polymers in equilibrium with an array of obstacles which models confinement in a gel or immersion in a melt. We find a crossover in both the geometrical and the topological behavior of the polymer. When the polymers' radius of gyration, RG, and that of the region containing the knot, R(G,k), are small compared to the distance b between the obstacles, the knot is weakly localize...

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 of intersections, defines the framing function of the knot. We show that the framing function is symmetric except at a finite number of points. The symmetry ...

The shape of a NURBS curve or patch is defined by the location of its control points, the weights associated with these points, and the parameter intervals, also called the knot vector. Most of the curve and patch design methods assume that the knot vector is constant and the user is allowed to modify only the control points and the weights. The possibility of controlling the shape through the ...

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...