# 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 planes tangent to the curve in more than one place?

• What precisely do you mean? You could compute it in a lot of standard ways, adapting Newton's method. Do you just want to see examples or are you primarily interested in computational heuristics? – Ryan Budney Dec 31 '13 at 20:07
• I'd be pretty happy with a decent picture, even. – Eric Zaslow Jan 2 '14 at 22:06
• Given a knot in $\mathbb R^3$ its set of tangent tangent lines is a 1-manifold in $\mathbb RP^2$, and generically this is an immersed circle. So the bitangent locus is the collection of double-points of this immersed curve in $\mathbb RP^2$. I think basically every null-homologous immersion $S^1 \to \mathbb RP^2$ is realizable as the tangent lines of a knot in $\mathbb RP^2$ so there's little restriction on the bitangent locus beyond that. – Ryan Budney Jan 2 '14 at 22:55

p.42: An aﬃne chart of the cone curve $C$ lying on a quadric cone $Q$ and its tangent developable $Z$.