4 research outputs found
On volume-preserving vector fields and finite type invariants of knots
We consider the general nonvanishing, divergence-free vector fields defined
on a domain in three space and tangent to its boundary. Based on the theory of
finite type invariants, we define a family of invariants for such fields, in
the style of Arnold's asymptotic linking number. Our approach is based on the
configuration space integrals due to Bott and Taubes.Comment: 30 pages, 6 figures, exposition improve
Associahedron, cyclohedron, and permutohedron as compactifications of configuration spaces
As in the case of the associahedron and cyclohedron, the permutohedron can
also be defined as an appropriate compactification of a configuration space of
points on an interval or on a circle. The construction of the compactification
endows the permutohedron with a projection to the cyclohedron, and the
cyclohedron with a projection to the associahedron. We show that the preimages
of any point via these projections might not be homeomorphic to (a cell
decomposition of) a disk, but are still contractible. We briefly explain an
application of this result to the study of knot spaces from the point of view
of the Goodwillie-Weiss manifold calculus.Comment: 27 pages The new version gives a more detailed exposition for the
projection from the cyclohedron to the associahedron as maps of
compactifications of configuration spaces. We also develop a similar picture
for the projection from the permutohedron to the cyclohedron/associahedro