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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.)

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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:

4 Illustrated 3D joint of two circle arcs.

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!

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