Narrow Results

The image of a polygonal knot K under a spherical inversion of ℝ³ ∪ ∞ is a simple closed curve made of arcs of circles, perhaps some line segments, having the same knot type as the mirror image of K. But suppose we reconnect the vertices of the inverted polygon with straight lines, making a new polygon . This may be a different knot type. For example, a certain 7-segment figure-eight knot can be transformed to a figure-eight knot, a trefoil, or an unknot, by selecting different inverting spheres. Which knot types can be obtained from a given original polygon K under this process? We show that for large n, most n-segment knot types cannot be reached from one initial n-segment polygon, using a single inversion or even the whole Möbius group.

The number of knot types is bounded by the number of complementary domains of a certain system of round 2-spheres in ℝ³. We show the number of domains is at most polynomial in the number of spheres, and the number of spheres is itself a polynomial function of the number of edges of the original polygon. In the analysis, we obtain an exact formula for the number of complementary domains of any collection of round 2-spheres in ℝ³. On the other hand, the number of knot types that can be represented by n-segment polygons is exponential in n.

Our construction can be interpreted as a particular instance of building polygonal knots in non-Euclidean metrics. In particular, start with a list of n vertices in ℝ³ and connect them with arcs of circles instead of line segments: Which knots can be obtained? Our polygonal inversion construction is equivalent to picking one fixed point p ∈ ℝ³ and replacing each edge of K by an arc of the circle determined by p and the endpoints of the edge.

The stick index of a knot is the least number of line segments required to build the knot in space. We define two analogous 2-dimensional invariants, the planar stick index, which is the least number of line segments in the plane to build a projection, and the spherical stick index, which is the least number of great circle arcs to build a projection on the sphere. We find bounds on these quantities in terms of other knot invariants, and give planar stick and spherical stick constructions for torus knots and for compositions of trefoils. In particular, unlike most knot invariants, we show that the spherical stick index distinguishes between the granny and square knots, and that composing a nontrivial knot with a second nontrivial knot need not increase its spherical stick index.

It is known that every nontrivial knot has at least two quadrisecants. Given a knot, we mark each intersection point of each of its quadrisecants. Replacing each subarc between two nearby marked points with a straight line segment joining them, we obtain a polygonal closed curve which we will call the quadrisecant approximation of the given knot. We show that for any hexagonal trefoil knot, there are only three quadrisecants, and the resulting quadrisecant approximation has the same knot type.

In 1983 Conway and Gordon proved that any embedding of the complete graph K₇ into ℝ³ contains at least one nontrivial knot as its Hamiltonian cycle. After their work knots (also links) are considered as intrinsic properties of abstract graphs, and numerous subsequent works have been continued until recently. In this paper, we are interested in knotted Hamiltonian cycles in linear embedding of K₇. Concretely it is shown that any linear embedding of K₇ contains at most three figure-8 knots.

We investigate a discrete version of the Möbius energy, that is of geometric interest in its own right and is defined on equilateral polygons with n segments. We show that the Γ-limit regarding L^q or W^1,q convergence, q ∈ [1, ∞] of these energies as n → ∞ is the smooth Möbius energy. This result directly implies the convergence of almost minimizers of the discrete energies to minimizers of the smooth energy if we can guarantee that the limit of the discrete curves belongs to the same knot class. Additionally, we show that the unique minimizer amongst all polygons is the regular n-gon. Moreover, discrete overall minimizers converge to the round circle.

A quadrisecant line of a knot $K$ is a straight line which intersects $K$ in four points, and a quadrisecant is a 4-tuple of points of $K$ which lie in order along the quadrisecant line. If $K$ has a finite number of quadrisecants, take $W$ to be the set of points of $K$ which are in a quadrisecant. Replace each subarc of $K$ between two adjacent points of $W$ along $K$ with the straight line segment between them. This gives the quadrisecant approximation of $K$. It was conjectured that the quadrisecant approximation is always a knot with the same knot type as the original knot. We show that every knot type contains two knots, the quadrisecant approximation of one knot has self-intersections while the quadrisecant approximation of the other knot is a knot with a different knot type.

For a polygon in the $d$-dimensional Euclidean space, we give two kinds of normalizations of its $m$th midpoint polygon by a homothetic transformation and an affine transformation, respectively. As $m$ goes to infinity, the normalizations will approach “regular” polygons inscribed in an ellipse and a generalized Lissajous curve, respectively, where the curves may be degenerate.

The most interesting case is when $d=3$, where polygons with all its $m$th midpoint polygons knotted are discovered and discussed. Such polygonal knots can be seen as a discrete version of the Lissajous knots.

The stick number and the edge length of a knot type in the simple hexagonal lattice (sh-lattice) are the minimal numbers of sticks and edges required, respectively, to construct a knot of the given type in sh-lattice. By introducing a linear transformation between lattices, we prove that for any given knot both values in the sh-lattice are strictly less than the values in the cubic lattice. Finally, we show that the only non-trivial $11$-stick knots in the sh-lattice are the trefoil knot ($3_1$) and the figure-eight knot ($4_1$).

The minimal number of straight line segments required to form a given knot or link in ℝ³ is determined for a family of torus knots and links.