14
votes
2answers
244 views

Random rings linked into one component?

Let $S$ be a sphere of unit radius. Let $C_n$ be a collection of unit-radius circles/rings whose centers are (uniformly distributed) random points in $S$, and which are oriented (tilted) randomly ...
2
votes
2answers
331 views

Infinite knot composed of parallel helices

I am wondering if there is a developed theory of "infinite knots" that could capture this object, and tell me something of its knot properties. Imagine vertical helices in $\mathbb{R}^3$, each ...
9
votes
3answers
448 views

Circle-arc number of a knot

I would like to build knots in $\mathbb{R}^3$ from arcs of unit-radius (planar) circles, joined together at points where the tangents match. Thus the knot will have curvature $1$ at all but the ...
5
votes
2answers
359 views

Tangled Knot Function

I am seeking a function $f: \mathbb{R}^3 \to \mathbb{R}^3$ that has these properties: (1) When iterated $n$ times starting from some $p$, connecting the points in order with segments and closing ...
2
votes
2answers
381 views

Is it possible to reliably generate a particular approximation of an ideal knot via a simulated annealing approach?

Say I take a cord, tie a loose knot in three-dimensional space, and pull tightly on the ends to generate an approximation of an ideal knot. If the cord has a fixed knot topology and a random initial ...
46
votes
4answers
3k views

Tying knots with reflecting lightrays

Let a lightray bounce around inside a cube whose faces are (internal) mirrors. If its slopes are rational, it will eventually form a cycle. For example, starting with a point $p_0$ in the interior of ...
6
votes
2answers
572 views

Lattice Stick Number vs. Stick Number of Knot

Can the lattice stick number of a knot be bounded by the stick number of the knot? The stick number $S(K)$ of a knot $K$ is the fewest number of segments needed to realize it by a simple 3D ...