1
vote
1answer
98 views

Bitangent locus of torus knots

Anyone know how to compute the bitangent locus of a space curve, e.g. a torus knot (pick whatever parametrization you like)? Specifically, what is the set of normal vectors (in the two-sphere) of ...
3
votes
1answer
110 views

Some questions about ideal knots

The ropelength of a knot curve $C$ is defined as the ratio $L(C) = \lambda(C)/ \tau(C)$, where $\lambda(C)$ is the length of $C$ and $\tau(C)$ is the thickness of the knot defined by $C$ [from ...
5
votes
1answer
307 views

Euclidean symmetries of torus links in R^3

I have a question about whether Ryan Budney's question: Torus knots in Euclidean space -- a symmetry argument can be extended to links. He asks: Suppose you have a $(p,q)$ torus knot $K$ in ...
1
vote
1answer
382 views

What is the shape of a tight open trefoil?

Take an infinitely long rope of diameter 1, ideally flexible and ideally slippery. Tie a trefoil knot into it and pull it tight. Describe the resulting rope shape analytically. The problem is ...
7
votes
2answers
1k views

Slice knots and exotic $\mathbb R^4$

In the http://arxiv.org/abs/math/0606464v1 I read "If you want to prove existence of exotic smooth structure on $\mathbb R^4$ you can do this if you are in possession of a knot which is ...
8
votes
1answer
717 views

The total curvature of very knotty knots

One of my favorite theorems is that of Fáry-Milnor, stating that the total curvature of a knot in $\mathbb R^3$ which is not an unknot (an ununknot) is at least $4\pi$. Can one quantify the way in ...