Yuanan Diao

On Random Knots

In this paper, we consider knotting of Gaussian random polygons in 3-space. A Gaussian random polygon is a piecewise linear circle with n edges in which the length of the edges follows a Gaussian distribution. We prove a continuum version of Kestens Pattern Theorem for these polygons, and use this to prove that the probability that a Gaussian random polygon of n edges in 3-space is knotted tends to one exponentially rapidly as n tends to infinity. We study the properties of Gaussian random knots, and prove that the entanglement complexity of Gaussian random knots gets arbitrarily large as n tends to infinity. We also prove that almost all Gaussian random knots are chiral.

MINIMAL KNOTTED POLYGONS ON THE CUBIC LATTICE

The polygons on the cubic-lattice have played an important role in simulating various circular molecules, especially the ones with relatively big volumes. There have been a lot of theoretical studies and computer simulations devoted to this subject. The questions are mostly around the knottedness of such a polygon, such as what kind of knots can appear in a polygon of given length, how often it can occur, etc. A very often asked and long standing question is about the minimal length of a knotted polygon. It is well-known that there are knotted polygons on the lattice with 24 steps yet it is unproved up to this date that 24 is the minimal number of steps needed. In this paper, we prove that in order to obtain a knotted polygon on the cubic lattice, at least 24 steps are needed and we can only have trefoils with 24 steps.

The complexity of lattice knots

Abstract A family of polygonal knots Kn on the cubical lattice is constructed with the property that the quotient of length L(Kn) over the crossing number Cr(Kn) approaches zero as L approaches infinity. More precisely Cr(K n ) = O(L(K n ) 4 3 ) . It is shown that this construction is optimal in the sense that for any knot K on the cubical lattice with length L and Cr crossings Cr ⩽ 3.2L 4 3 .

The existence of subspace wavelet sets

Let H be a reducing subspace of L2(Rd), that is, a closed subspace of L2(Rd) with the property that f(Amt - l) ∈ H for any f ∈ H, m ∈ Z and l ∈ Zd, where A is a d × d expansive matrix. It is known that H is a reducing subspace if and only if there exists a measurable subset M of Rd such that AtM = M and F(H) = L2(Rd) ċ χM. Under some given conditions of M, it is known that there exist A-dilation subspace wavelet sets iwith respect to H. In this paper, we prove that this holds in general.

The average crossing number of equilateral random polygons

In this paper, we study the average crossing number of equilateral random walks and polygons. We show that the mean average crossing number ACN of all equilateral random walks of length n is of the form . A similar result holds for equilateral random polygons. These results are confirmed by our numerical studies. Furthermore, our numerical studies indicate that when random polygons of length n are divided into individual knot types, the for each knot type can be described by a function of the form where a, b and c are constants depending on and n0 is the minimal number of segments required to form . The profiles diverge from each other, with more complex knots showing higher than less complex knots. Moreover, the profiles intersect with the ACN profile of all closed walks. These points of intersection define the equilibrium length of , i.e., the chain length at which a statistical ensemble of configurations with given knot type —upon cutting, equilibration and reclosure to a new knot type —does not show a tendency to increase or decrease . This concept of equilibrium length seems to be universal, and applies also to other length-dependent observables for random knots, such as the mean radius of gyration Rg.

Thicknesses of knots

In this paper we define a set of radii called thickness for simple closed curves denoted by K , which are assumed to be differentiable. These radii capture a balanced view between the geometric and the topological properties of these curves. One can think of these radii as representing the thickness of a rope in space and of K as the core of the rope. Great care is taken to define our radii in order to gain freedom from small pieces with large curvature in the curve. Intuitively, this means that we tend to allow the surface of the ropes that represent the knots to deform into a non smooth surface. But as long as the radius of the rope is less than the thickness so defined, the surface of the rope will remain a two manifold and the rope (as a solid torus) can be deformed onto K via strong deformation retract. In this paper we explore basic properties of these thicknesses and discuss the relationship amongst them.

Subspaces with normalized tight frame wavelets in R

In this paper we investigate the subspaces of L 2 (R) which have normalized tight frame wavelets that are defined by set functions on some measurable subsets of R called Bessel sets. We show that a subspace admitting such a normalized tight frame wavelet falls into a class of subspaces called reducing subspaces. We also consider the subspaces of L 2 (R) that are generated by a Bessel set E in a special way. We present some results concerning the relation between a Bessel set E and the corresponding subspace of L 2 (R) which either has a normalized tight frame wavelet defined by the set function on E or is generated by E.

Bounds for the minimum step number of knots in the simple cubic lattice

Knots are found in DNA as well as in proteins, and they have been shown to be good tools for structural analysis of these molecules. An important parameter to consider in the artificial construction of these molecules is the minimum number of monomers needed to make a knot. Here we address this problem by characterizing, both analytically and numerically, the minimum length (also called minimum step number) needed to form a particular knot in the simple cubic lattice. Our analytical work is based on improvement of a method introduced by Diao to enumerate conformations of a given knot type for a fixed length. This method allows us to extend the previously known result on the minimum step number of the trefoil knot 31 (which is 24) to the knots 41 and 51 and show that the minimum step numbers for the 41 and 51 knots are 30 and 34, respectively. Using an independent method based on the BFACF algorithm, we provide a complete list of numerical estimates (upper bounds) of the minimum step numbers for prime knots up to ten crossings, which are improvements over current published numerical results. We enumerate all minimum lattice knots of a given type and partition them into classes defined by BFACF type 0 moves.

THE LOWER BOUNDS OF THE LENGTHS OF THICK KNOTS

In this paper, we derive a formula that provides lower bounds on the minimal arc length required to tie a unit thickness knot in terms of the minimal crossing number of the knot. We prove that for any nontrivial knot with unit thickness, the minimal arc length is at least 24. This answers an open question (posted in [13]) in the negative: one cannot tie a knot using one foot-length of one-inch (diameter) rope. For knots with minimal crossing numbers up to 1850, our result yields larger lower bounds on the lengths of the knots than the known results.

The number of smallest knots on the cubic lattice

It has been shown that the smallest knots on the cubic lattice are all trefoils of length 24. In this paper, we show that the number of such unrooted knots on the cubic lattice is 3496.