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Let $\gamma$ be a smooth, closed, unknotted curve embedded in $\mathbb{R}^3$.

Q. Does there always exist a smooth, embedded, genus-zero surface $S \subset \mathbb{R}^3$ such that $\gamma$ is a (closed) geodesic on $S$?

Here the metric on $S$ is inherited from $\mathbb{R}^3$. The curve $\gamma$ could be knotted, but it is non-self-intersecting. I am seeking $S$ homeomorphic to a sphere, i.e., genus-zero.

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          ![SpaceCurve][1]

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One can construct an appropriate surface patch locally in a neighborhood of each point $x \in \gamma$, but it is unclear to me how to argue that these patches can be completed to a genus-$0$ embedded surface $S$.

Revision. Andy Putnam and Igor Rivin both answered the original question: No if $\gamma$ is knotted. So I have revised the question to restrict $\gamma$ to be unknotted.

Let $\gamma$ be a smooth, closed, unknotted curve embedded in $\mathbb{R}^3$.

Q. Does there always exist a smooth, embedded, genus-zero surface $S \subset \mathbb{R}^3$ such that $\gamma$ is a (closed) geodesic on $S$?

Here the metric on $S$ is inherited from $\mathbb{R}^3$. The curve $\gamma$ could be knotted, but it is non-self-intersecting. I am seeking $S$ homeomorphic to a sphere, i.e., genus-zero.

---

          ![SpaceCurve][1]

---

One can construct an appropriate surface patch locally in a neighborhood of each point $x \in \gamma$, but it is unclear to me how to argue that these patches can be completed to a genus-$0$ embedded surface $S$.

Revision. Andy Putman and Igor Rivin both answered the original question: No if $\gamma$ is knotted. So I have revised the question to restrict $\gamma$ to be unknotted.

Let $\gamma$ be a smooth, closed, unknotted curve embedded in $\mathbb{R}^3$.

Q. Does there always exist a smooth, embedded, genus-zero surface $S \subset \mathbb{R}^3$ such that $\gamma$ is a (closed) geodesic on $S$?

Here the metric on $S$ is inherited from $\mathbb{R}^3$. The curve $\gamma$ could be knotted, but it is non-self-intersecting. I am seeking $S$ homeomorphic to a sphere, i.e., genus-zero.

---

          ![SpaceCurve][1]

---

One can construct an appropriate surface patch locally in a neighborhood of each point $x \in \gamma$, but it is unclear to me how to argue that these patches can be completed to a genus-$0$ embedded surface $S$.

Revision. Andy Putnam and Igor Rivin both answered the question:  No if $\gamma$ is knotted. So I have revised the question to restrict $\gamma$ to be unknotted.

Let $\gamma$ be a smooth, closed, unknotted curve embedded in $\mathbb{R}^3$.

Q. Does there always exist a smooth, embedded, genus-zero surface $S \subset \mathbb{R}^3$ such that $\gamma$ is a (closed) geodesic on $S$?

Here the metric on $S$ is inherited from $\mathbb{R}^3$. The curve $\gamma$ could be knotted, but it is non-self-intersecting. I am seeking $S$ homeomorphic to a sphere, i.e., genus-zero.

---

          ![SpaceCurve][1]

---

One can construct an appropriate surface patch locally in a neighborhood of each point $x \in \gamma$, but it is unclear to me how to argue that these patches can be completed to a genus-$0$ embedded surface $S$.

Revision. Andy Putnam and Igor Rivin both answered the original question:  No if $\gamma$ is knotted. So I have revised the question to restrict $\gamma$ to be unknotted.

Let $\gamma$ be a smooth, closed curve embedded in $\mathbb{R}^3$.

Q. Does there always exist a smooth, embedded, genus-zero surface $S \subset \mathbb{R}^3$ such that $\gamma$ is a (closed) geodesic on $S$?

Here the metric on $S$ is inherited from $\mathbb{R}^3$. The curve $\gamma$ could be knotted, but it is non-self-intersecting. I am seeking $S$ homeomorphic to a sphere, i.e., genus-zero.

---

          ![SpaceCurve][1]

---

One can construct an appropriate surface patch locally in a neighborhood of each point $x \in \gamma$, but it is unclear to me how to argue that these patches can be completed to a genus-$0$ embedded surface $S$.

Let $\gamma$ be a smooth, closed, unknotted curve embedded in $\mathbb{R}^3$.

Q. Does there always exist a smooth, embedded, genus-zero surface $S \subset \mathbb{R}^3$ such that $\gamma$ is a (closed) geodesic on $S$?

Here the metric on $S$ is inherited from $\mathbb{R}^3$. The curve $\gamma$ could be knotted, but it is non-self-intersecting. I am seeking $S$ homeomorphic to a sphere, i.e., genus-zero.

---

          ![SpaceCurve][1]

---

One can construct an appropriate surface patch locally in a neighborhood of each point $x \in \gamma$, but it is unclear to me how to argue that these patches can be completed to a genus-$0$ embedded surface $S$.

Revision. Andy Putnam and Igor Rivin both answered the question: No if $\gamma$ is knotted. So I have revised the question to restrict $\gamma$ to be unknotted.