I would like to build knots in $\mathbb{R}^3$ from arcs
of unit-radius (planar) circles, joined together at points where
the tangents match. Thus the knot will have
curvature $1$ at all but the joints.
Here is an example of how two arcs might join:
Define the circle-arc number $C(K)$ of a knot $K$ as the fewest number of such arcs from which one can build a nonselfinterecting curve in space representing $K$. This number is analogous to the stick number of a knot, except that the pieces are arcs, and there is a tangent-joining condition.
I would be interested to learn of bounds on $C(K)$ in terms of other knot quantities, for example, the stick number, or the crossing number cr$(K)$.
Here is an example of what I have in mind.
It appears that one might be able to build a trefoil from six
arcs, something like this:
However, the above picture is actually planar, and I have not
verified carefully that this is achievable in $\mathbb{R}^3$!
Has this concept been studied before? If so, pointers would be welcomed. Thanks!
Addendum. The trefoil can be realized with six arcs:
(The black triangle vertices indicate the circle centers on the plane before their arcs are twisted into 3D.)
