Questions tagged [stick-knots]
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17 questions
6
votes
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answers
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Is the unit-stick number of a knot equal to its stick number?
Define the unit-stick number $\sigma_1(K)$ of a knot $K$ to be the fewest unit-length
sticks that can realize $K$.
Clearly $\sigma_1(K)$ is at least the stick number $\sigma(K)$.
It is known that the ...
- 151k
23
votes
1
answer
524
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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 ...
- 151k
4
votes
1
answer
263
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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 $\mathbb{R}^3$.
...
- 151k
7
votes
2
answers
259
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 ...
- 151k
10
votes
3
answers
950
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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 ...
- 151k
5
votes
4
answers
660
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.)...
- 51
3
votes
1
answer
394
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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 ...
- 151k
27
votes
1
answer
2k
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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 ...
- 5,191
7
votes
2
answers
539
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 ...
- 151k
8
votes
3
answers
793
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 $n$ ...
- 436
4
votes
2
answers
1k
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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 ...
- 151k
9
votes
1
answer
1k
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....
- 921
30
votes
3
answers
2k
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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 ...
- 13.1k
26
votes
5
answers
2k
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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 number
(the minimal number of ...
- 151k
7
votes
1
answer
938
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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 ...
- 151k
8
votes
2
answers
741
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....
- 151k
16
votes
2
answers
489
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 ...
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