  Let $f:S^1\to R$ be a generic map. We may use $f$ to define a new map
$\tilde{f}:S^1\to R^3$ by $\tilde{f}(t) = (-f(t),f'(t),-f''(t))$, and if $f$ is
an embedding then the image of $\tilde{f}$ will be a knot. Knots defined by
such parametrizations are called holonomic knots. They were introduced in 1997
by Vassiliev, who proved that every knot type can be represented by a holonomic
knot. Our main result is that any two holonomic knots which represent the same
knot type are isotopic in the space of holonomic knots. A second result emerges
through the techniques used to prove the main result: strong and unexpected
connections between the topology of knots and the algebraic solution to the
conjugacy problem in the braid groups, via the work of Garside. We also discuss
related parametrizations of Legendrian knots, and uncover connections between
the concepts of holonomic and Legendrian parametrizations of knots.
