Related papers: Vertex distortion of lattice knots

For a connected cobordism S between two knots K1,K2 in S3, we establish an inequality involving the number of local maxima, the genus of S, and the torsion orders of Kht(K1),Kht(K2), where Kht denotes Lee's perturbation of Khovanov…

The (isothermic) compressibility of lattice knots can be examined as a model of the effects of topology and geometry on the compressibility of ring polymers. In this paper, the compressibility of minimal length lattice knots in the simple…

In 2008, Vallentin made a conjecture involving the least distortion of an embedding of a distance-regular graph into Euclidean space. Vallentin's conjecture implies that for a least distortion Euclidean embedding of a distance-regular graph…

A knotted ribbon is one of physical aspect of a knot. A folded ribbon knot is a depiction of a knot obtained by folding a long and thin rectangular strip to become flat. The ribbonlength of a knot type can be defined as the minimum length…

We introduce two numerical invariants, the waist and the trunk of knots. The waist of a closed incompressible surface in the complement of a knot is defined as the minimal intersection number of all compressing disks for the surface in the…

The warping degree of an oriented knot diagram is the minimal number of crossing changes which are required to obtain a monotone knot diagram from the diagram. The minimal warping degree of a knot is the minimal value of the warping degree…

We calculate mean square deviations for crystals in one and two dimensions. For the two dimensional lattices, we consider several distinct geometries (i.e. square, triangular, and honeycomb), and we find the same essential phenomena for…

Connes' distance formula is applied to endow linear metric to three 1D lattices of different topology, with a generalization of lattice Dirac operator written down by Dimakis et al to contain a non-unitary link-variable. Geometric…

Traditional Cauchy-Stokes decomposition of velocity gradient tensor gives a symmetric and an anti-symmetric subtensors which are called the strain-rate and vorticity tensors. There are two problems with Cauchy-Stokes decomposition. The…

With increasing applied current we show that the moving vortex lattice changes its structure from a triangular one to a set of parallel vortex rows in a pinning free superconductor. This effect originates from the change of the shape of the…

If a knot has the Alexander polynomial not equal to 1, then it is linear $n$-colorable. By means of such a coloring, such a knot is given an upper bound for the minimal quandle order, i.e., the minimal order of a quandle with which the knot…

In the following paper we continue the work of Bimonte-Lizzi-Sparano on distances on a one dimensional lattice. We succeed in proving analytically the exact formulae for such distances. We find that the distance to an even point on the…

One important question in the theory of lattices is to detect a shortest vector: given a norm and a lattice, what is the smallest norm attained by a non-zero vector contained in the lattice? We focus on the infinity norm and work with…

We describe the geometry of bend distortions in twist-bend nematic liquid crystals in terms of their fundamental degeneracies, which we call $\beta$ lines. These represent a new class of line-like topological defect. We use them to…

This paper is expository and is accessible to students. We define simple invariants of knots or links (linking number, Arf-Casson invariants and Alexander-Conway polynomials) motivated by interesting results whose statements are accessible…

Relative self-linking and linking "numbers" for pairs of knots in oriented 3-manifolds are defined in terms of intersection invariants of immersed surfaces in 4-manifolds. The resulting concordance invariants generalize the usual…

We determine the locally flat cobordism distance between torus knots with small and large braid index, up to high precision. Here small means 2, 3, 4, or 6. As an application, we derive a surprising fact about torus knots that appear as…

Twisted torus knots and links are given by twisting adjacent strands of a torus link. They are geometrically simple and contain many examples of the smallest volume hyperbolic knots. Many are also Lorenz links. We study the geometry of…

The ropelength of a knot or link is the minimal number of inches of 1-inch-thick rope that it takes to tie it. The relationship of this measurement to knot and link invariants has been studied by various authors. We give the first results…

The ropelength of a knot is the quotient of its length and its thickness, the radius of the largest embedded normal tube around the knot. We prove existence and regularity for ropelength minimizers in any knot or link type; these are…