Résume | An algorithm which takes any knot to its "normal form" will be described and
computer animations showing the isotopy that does this will be demonstrated.
The algorithm is a discrete version of gradient descent along the sum A=E+R
of two functionals: the Euler functional E (the integral over the curve of the
square of its curvature) and a simple repulsive functional R that grows very
rapidly when points that are not close along the curve become close to each
other in space (thus R makes crossing changes impossible). The results of
computer experiments with our software will be described: they showed that
there is a unique (up to isometry) normal form (i.e., a curve at which A yields
the minimum for over curves in the same isotopy class) for all knots with seven
crossings or less. Unfortunately, our algorithm does not always take the knot
to its normal form, it sometimes stops at a local minimum of A, in particular,
it does not always unravel certain versions of the unknot. On the other hand,
our algorithm turned out to be an excellent mathematical model of physical
experiments with knots made of flexible, resilient, but unstretchable wire
(a few such experiments will be demonstrated). The talk is based on joint
work with S.Avvakumov and O.Karpenkov. |