Unknotted Curves Research Articles

Unknotted curves on genus-one Seifert surfaces of Whitehead doubles

Unknotted curves on genus-one Seifert surfaces of Whitehead doubles

Cuts for 3-D magnetic scalar potentials: Visualizing unintuitive surfaces arising from trivial knots

A wealth of literature exists on computing and visualizing cuts for the magnetic scalar potential of a current carrying conductor via Finite Element Methods (FEM) and harmonic maps to the circle. By a cut we refer to an orientable surface bounded by a given current carrying path (such that the flux through it may be computed) that restricts contour integrals on a curl-zero vector field to those that do not link the current-carrying path, analogous to branch cuts of complex analysis. This work is concerned with a study of a peculiar contour that illustrates topologically unintuitive aspects of cuts obtained from a trivial loop and raises questions about the notion of an optimal cut. Specifically, an unknotted curve that bounds only high genus surfaces in its convex hull is analyzed. The current work considers the geometric realization as a current-carrying wire in order to construct a magnetic scalar potential. Moreover, we consider the problem of choosing an energy functional on the space of maps, suggesting an algorithm for computing cuts via minimizing a conformally invariant functional utilizing Newton iteration.

Movie moves for singular link cobordisms in 4-dimensional space

Two singular links are cobordant if one can be obtained from the other by singular link isotopy together with a combination of births or deaths of simple unknotted curves, and saddle point transformations. A movie description of a singular link cobordism in 4-space is a sequence of singular link diagrams obtained from a projection of the cobordism into 3-space by taking 2-dimensional cross-sections perpendicular to a fixed direction. We present a set of movie moves that are sufficient to connect any two movies of isotopic singular link cobordisms.

Geometric Knot Theory

Geometric knot theory studies relations between geometric properties of a space curve and the knot type it represents. As examples, knotted curves have quadrisecant lines, and have more distortion and more total curvature than (some) unknotted curves. Geometric energies for space curves – like the Möbius energy, ropelength and various regularizations – can be minimized within a given knot type to give an optimal shape for the knot. Increasing interest in this area over the past decade is partly due to various applications, for instance to random knots and polymers, to topological fluid dynamics and to the molecular biology of DNA. This workshop focused on the mathematics behind these applications, drawing on techniques from algebraic topology, differential geometry, integral geometry, geometric measure theory, calculus of variations, nonlinear optimization and harmonic analysis.

On multiply twisted knots that are Seifert fibered or toroidal

We consider knots whose diagrams have a high amount of twisting of multiple strands. By encircling twists on multiple strands with unknotted curves, we obtain a link called a generalized augmented link. Dehn filling this link gives the original knot. We classify those generalized augmented links that are Seifert fibered, and give a torus decomposition for those that are toroidal. In particular, we find that each component of the torus decomposition is either “trivial”, in some sense, or homeomorphic to the complement of a generalized augmented link. We show this structure persists under high Dehn filling, giving results on the torus decomposition of knots with generalized twist regions and a high amount of twisting. As an application, we give lower bounds on the Gromov norms of these knot complements and of generalized

Area Inequalities for Embedded Disks Spanning Unknotted Curves

We show that a smooth unknotted curve in ℝ3 satisfies an isoperimetric inequality that bounds the area of an embedded disk spanning the curve in terms of two parameters: the length L of the curve, and the thickness r (maximal radius of an embedded tubular neighborhood) of the curve. For fixed length, the expression giving the upper bound on the area grows exponentially in 1/r 2. In the direction of lower bounds, we give a sequence of length one curves with r → 0 for which the area of any spanning disk is bounded from below by a function that grows exponentially with 1/r. In particular, given any constant A, there is a smooth, unknotted length one curve for which the area of a smallest embedded spanning disk is greater than A.

ON MARKOV'S THEOREM

Let χ be an oriented link type in the oriented 3-sphere S3 or ℝ3 = S3 - {∞}. A representative X ∈ χ is said to be a closed braid if there is an unknotted curve A ⊂ S3 - X (the axis) and a choice of fibration ℋ of the open solid torus S3 - A by meridian discs {Hθ : θ ∈ [0, 2 π]}, such that whenever X meets a fiber Hθ the intersection is transverse. Closed braid representations of χ are not unique, and Markov's well-known theorem asserts that any two are related by a finite sequence of elementary moves. The main result in this paper is to give a new proof of Markov's theorem. We hope that our new proof will be of interest because it gives new insight into the geometry.

Examples of Unknotted Curves Which Bound Only Surfaces of High Genus Within Their Convex Hulls

Examples of Unknotted Curves Which Bound Only Surfaces of High Genus Within Their Convex Hulls

Certain Multiple Valued Harmonic Function

The difference equation $v({x^ + }) - v(x) = u(x)$ is solved for any harmonic $u$ in the covering space of an unknotted curve.

Certain multiple valued harmonic function

The difference equation v ( x + ) − v ( x ) = u ( x ) v({x^ + }) - v(x) = u(x) is solved for any harmonic u u in the covering space of an unknotted curve.

Errata: A Class of Unknotted Curves in 3-Space

Errata: A Class of Unknotted Curves in 3-Space

A Class of Unknotted Curves in 3-Space

A Class of Unknotted Curves in 3-Space

A class of unknotted curves in 3-space

A class of unknotted curves in 3-space

A Remarkable Simple Closed Curve

Two simple closed curves in ordinary three dimensional space R are called equivalent' if there is an orientation-preserving homeomorphism of R on itself which transforms one curve into the other. A curve is called unknotted if it is equivalent to the circle X2 + y2 = 1, Z = 0, otherwise knotted. In a recent paper2 we have introduced the terms tame, for a curve equivalent to a polygon, and wild, for one not equivalent to a polygon 3. It is clear that an unknotted curve is tame and has an abelian (hence infinite cyclic) group. Whether conversely a curve which is tame and has an abelian group is necessarily unknotted is still an open question 4. On the other hand a wild (and hence knotted) curve whose group is abelian has been constructed 5. It is known that a simple closed curve in R is unknotted if and only if it is equivalent to a curve F which is a subset of a plane in R. Let us call a simple closed curve almost unknotted if it is equivalent to a curve which has the following property: (*) For any neighborhood U of a given p there is a neighborhood V C U of p and a homeomorphism so of R on itself such that (i) jp(q) = q for every q e V, and (ii) p(F V) is a subset of a plane in R. There exists a simple closed curve whose group is non-abelian although it is almost unknotted 6. A regular normed projection of this simple closed curve I is shown in figure 1. (Using the approach of F A one could easily construct a precise description). It has one singular point p; by applying the necessary condition developed in F A example 1.2 to a subarc of F of which p is the left endpoint it can be shown that F is wild 7. The group 8 of F is generated by elements an , bn X c, (n > 0) indicated in the usual way in figure 1. A set of defining relations is