Related papers: Knotting Probability of Equilateral Hexagons
We prove that every unit area convex pentagon is contained in a convex quadrilateral of area no greater than $3/\sqrt{5}$, and that every unit area convex hexagon is contained in a convex pentagon of area no greater than $7/6$. Both results…
In this paper we calculate the Ehrhart's polynomial associated with a 2-dimensional regular polytope (i.e. equilateral triangles) in $\mathbb Z^3$. The polynomial takes a relatively simple form in terms of the coordinates of the vertices of…
The study of (minimally) rigid graphs is motivated by numerous applications, mostly in robotics and bioinformatics. A major open problem concerns the number of embeddings of such graphs, up to rigid motions, in Euclidean space. We capture…
In this paper we study some aspects of knots and links in lens spaces. Namely, if we consider lens spaces as quotient of the unit ball $B^{3}$ with suitable identification of boundary points, then we can project the links on the equatorial…
We classify nonnegatively curved simply connected 4-manifolds with circle symmetry up to equivariant diffeomorphisms. The main problem is rule out knotted curves in the singular set of the orbit space. As an extension of this work we…
It has been conjectured by Rovelli that there is a correspondence between the space of link classes of a Riemannian 3-manifold and the space of 3-geometries (on the same manifold). An exact statement of his conjecture will be established…
We partially determine grid homology (combinatorial knot Floer homology) of diagonal knots, which are conjectured to be equivalent to positive braid knots, by exploiting nice grid diagrams. Its next-to-top term detects the number of prime…
In any dimension $2n \ge 6$ we show that certain spaces of symplectic embeddings of a polydisk into a product $B^4 \times \Bbb R^{2(n-2)}$ of a $4$-ball and Euclidean space, are not path connected. We also show that any pair of such…
We construct knots in S^3 with Heegaard splittings of arbitrarily high distance, in any genus. As an application, for any positive integers t and b we find a tunnel number t knot in the three-sphere which has no (t,b)-decomposition.
We discuss the probability of random knotting for a model of self-avoiding polygons whose segments are given by cylinders of unit length with radius $r$. We show numerically that the characteristic length of random knotting is roughly…
The knotting probability is defined by the probability with which an $N$-step self-avoiding polygon (SAP) with a fixed type of knot appears in the configuration space. We evaluate these probabilities for some knot types on a simple cubic…
We study the existence of equilateral polygons in planar integer lattices. Maehara showed that it's sufficient to work with rectangular lattices $\Lambda(m) = L[(1,0),(0,\sqrt{m})]$ with $m \equiv 3 \pmod{4}$. Building on results of Maehara…
We study surface knots in 4-space by using generic planar projections. These projections have fold points and cusps as their singularities and the image of the singular point set divides the plane into several regions. The width (or the…
A graph is called (generically) rigid in $\mathbb{R}^d$ if, for any choice of sufficiently generic edge lengths, it can be embedded in $\mathbb{R}^d$ in a finite number of distinct ways, modulo rigid transformations. Here we deal with the…
The inclusion of the space of all knots of a prescribed writhe in a particular isotopy class into the space of all knots in that isotopy class is a weak homotopy equivalence.
We give an infinite family of knots such that for any given $r \geq 3$, the family contains a knot which can be embedded on a hexagonal $r$-mosaic, but cannot fit on a hexagonal $r$-mosaic in an embedding that achieves its crossing number.…
We present experimental results on knotting in off-lattice self-avoiding polygons in the bead-chain model. Using Clisby's tree data structure and the scale-free pivot algorithm, for each $k$ between $10$ and $27$ we generated $2^{43-k}$…
Given a hyperbolic 3-manifold with torus boundary, we bound the change in volume under a Dehn filling where all slopes have length at least 2\pi. This result is applied to give explicit diagrammatic bounds on the volumes of many knots and…
We introduce and study knots and links in 2-dimensional complexes. In particular, we define linking numbers for oriented two-component links in 2-complexes and a Kauffman-type bracket polynomial for links in 2-complexes. We also discuss…
We compute the cohomology of polygon spaces using their identification to (semi) stable configuration of weighted points on complex projective line. This cohomology is already given by J.C.Hausmann and A. Knutson but we use a different…