Narrow Results

It is shown that Legendrian (respectively transverse) cable links in S³ with its standard tight contact structure, i.e. links consisting of an unknot and a cable of that unknot, are classified by their oriented link type and the classical invariants (Thurston–Bennequin invariant and rotation number in the Legendrian case, self-linking number in the transverse case). The analogous result is proved for torus knots in the 1-jet space J¹(S¹) with its standard tight contact structure.

Traynor ([11]) has described an example of a two-component Legendrian "circular helix link" Λ₀ ⊔ Λ₁ in the 1-jet space J¹(S¹) of the circle (with its canonical contact structure) that is topologically but not Legendrian isotopic to the link Λ₁ ⊔ Λ₀. We give a complete classification of the Legendrian realizations of this topological link type, as well as all other "cable links" in J¹(S¹).

The Wiener measure induces a measure of closed, convex, d – 1-dimensional, Euclidean (hyper-)surfaces that are the convex hulls of closed d-dimensional Brownian bridges. I present arguments and numerical evidence that this measure, for odd d, is generated by a local classical action of length dimension 2 that depends on geometric invariants of the d – 1-dimensional surface only.