Questions tagged [stick-knots]

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22
votes
1answer
469 views

Tying knots via gravity-assisted spaceship trajectories

Q. Can every knot be realized as the trajectory of a spaceship weaving among a finite number of fixed planets, subject to gravity alone?           To make this more ...
4
votes
1answer
220 views

Knotted TSP tours in 3D?

In the plane, the Euclidean TSP tour never crosses itself—it is always a simple polygon. I am wondering if there is a similar constraint for the Euclidean TSP tour of points in $\mathbb{R}^3$. ...
7
votes
2answers
233 views

Can every large point set be connected to a given knot?

Let $K$ be a given knot, and $P$ a set of points in $\mathbb{R}^3$ in general position, general position in the sense that no three points are collinear and no four coplanar. Define the point-set ...
10
votes
3answers
601 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 ...
4
votes
4answers
548 views

Stick knot questions: simple but may not be easy

I have a few questions about nonplanar "stick circuits" (or hexagons and higher $n$-gons) that you might be able to help with: (I know that $n=6$ is the minimum number of points to form a stick knot.)...
3
votes
1answer
325 views

Min Bend Orthogonal Knots

I am seeking literature on 3D orthogonal drawings of knots, especially minimum bend drawings. An orthogonal drawing employs segments parallel to the axes of a Cartesian coordinate system. A bend is a ...
27
votes
1answer
2k views

Does this knot invariant distinguish trefoil chiralities?

Let $C_N$ denote the labelled configuration of $N^{th}$ roots of unity with $p_J = e^{\frac{2\pi iJ}{N}}$ for $J = 1\ldots N$. As a corollary of something else I was playing around with, I recently ...
6
votes
2answers
481 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 ...
8
votes
3answers
721 views

Efficient topological triangulations of non-convex polyhedra

Does every polyhedron in $\mathbb{R}^3$ with $n$ triangular facets have a topological triangulation with complexity $O(n)$? Suppose $P$ is a non-convex polyhedron in $\mathbb{R}^3$ with $n$ ...
4
votes
2answers
850 views

Unknotting knots in 4D

Suppose one has a knot $K$ embedded in $\mathbb{R}^3$; but view $\mathbb{R}^3$ as a 3-flat in $\mathbb{R}^4$. Of course $K$ is not a knot in $\mathbb{R}^4$. I am wondering if there has been any study ...
9
votes
1answer
846 views

What are the statistics of prime knots in 3d Random walk?

This question on physics stackexchange https://physics.stackexchange.com/questions/12973/the-entropic-cost-of-tying-knots-in-polymers has a formulation which is perhaps more appropriate for this forum....
30
votes
3answers
1k views

Random knot on six vertices

This question is inspired by Joseph O'Rourke's beautiful question on random knots. Choose an random ordered 6-tuple of points on the unit sphere in $\mathbf{R}^3$, and form a knot by connecting ...
26
votes
5answers
1k views

Complexity of random knot with vertices on sphere

Connect $n$ random points on a sphere in a cycle of segments between succesive points:       I would like to know the growth rate, with respect to $n$, of the crossing ...
7
votes
1answer
784 views

Which knots' stick numbers are twice their crossing numbers?

Looking at a table of minimum stick numbers for knots (table here), it seems the known upper bound of $2 c(K)$ in terms of the knot crossing number $c(K)$ is realized by the trefoil $3_1$—it ...
7
votes
2answers
663 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 polygon....
16
votes
2answers
461 views

Are there piecewise-linear unknots that are not metrically unknottable?

A stick knot is a just a piecewise linear knot. We could define "stick isotopy" as isotopy that preserves the length of each linear piece. Are there stick knots which are topologically trival, but ...