Inscribing smooth knots with regular polygons

We study collections of planar curves that yield diagrams for all knots. In particular, we show that a very special class called potholder curves carries all knots. This has implications for realizing all knots and links as special types of meanders and braids. We also introduce and apply a method to compare the efficiency of various classes of curves that represent all knots.

This paper introduces the concept of a Fourier knot. A Fourier knot is a knot that is represented by a parametrized curve in three dimensional space such that the coordinate functions are finite Fourier series in the parameter. The previously studied Lissajous knots constitute a case of Fourier knots with single frequencies in each coordinate direction. Not all knots are Lissajous. In fact the trefoil knot and the figure eight knot are the first examples of non- Lissajous kno...

The goal of this paper is to discuss the possibility of finding an algorithm that can give all distinct knots up to a desired complexity. Two such algorithms are presented, one based on projections on a plane, the other on closed self-avoiding walks.

In this paper we summarise the work discussed in Ref. [1] and [2] (q-alg/9505003), in which we introduced a method helpful in solving the problem of knot classification. We also present results obtained since then.

A Chebyshev knot is a knot which admits a parametrization of the form $ x(t)=T_a(t); \ y(t)=T_b(t) ; \ z(t)= T_c(t + \phi), $ where $a,b,c$ are pairwise coprime, $T_n(t)$ is the Chebyshev polynomial of degree $n,$ and $\phi \in \RR .$ Chebyshev knots are non compact analogues of the classical Lissajous knots. We show that there are infinitely many Chebyshev knots with $\phi = 0.$ We also show that every knot is a Chebyshev knot.

We show that there exist non-trivial piecewise-linear (PL) knots with isolated singularities $S^{n-2}\subset S^n$, $n\geq 5$, whose complements have the homotopy type of a circle. This is in contrast to the case of smooth, PL locally-flat, and topological locally-flat knots, for which it is known that if the complement has the homotopy type of a circle, then the knot is trivial.

A ribbon is, intuitively, a smooth mapping of an annulus $S^1 \times I$ in 3-space having constant width $\varepsilon$. This can be formalized as a triple $(x,\varepsilon, \mathbf{u})$ where $x$ is smooth curve in 3-space and $\mathbf{u}$ is a unit vector field based along $x$. In the 1960s and 1970s, G. Calugareanu, G. H. White, and F. B. Fuller proved relationships between the geometry and topology of thin ribbons, in particular the "Link = Twist + Writhe" theorem that has ...

Early last century witnessed both the complete classification of 2-dimensional manifolds and a proof that classification of 4-dimensional manifolds is undecidable, setting up 3-dimensional manifolds as a central battleground of topology to this day. A rather important subset of the 3-manifolds has turned out to be the knotspaces, the manifolds left when a thin tube around a knot in 3D space is excised. Given a knot diagram it would be desirable to provide as compact a descrip...

This paper, to be regularly updated, lists those prime knots with the fewest possible number of crossings for which values of basic knot invariants, such as the unknotting number or the smooth 4-genus, are unknown. This list is being developed in conjunction with "KnotInfo" (www.indiana.edu/~knotinfo), a web-based table of knot invariants.

A physical interpretation of the rope simulated by the SONO algorithm is presented. Properties of the tight polygonal knots delivered by the algorithm are analyzed. An algorithm for bounding the ropelength of a smooth inscribed knot is shown. Two ways of calculating the ropelength of tight polygonal knots are compared. An analytical calculation performed for a model knot shows that an appropriately weighted average should provide a good estimation of the minimum ropelength fo...